What we are beginning to see is that that homicide increase that began last year, it has not disappeared.

虽然问题是关于这三个that，但从提问来看，我觉得你能理解他们的含义，之所以有点自我怀疑，是因为不了解左偏置结构：

that homicide increase that began last year, it has not disappeared.

=that homicide increase that began last year has not disappeared.

句子中的一个名词短语，有可能会被提到句子结构之外，而在原位留下一个代词和这个名词短语互指。名词短语提到句首的，因为处于主句的左边，所以叫做左偏置结构left dislocation。 相应的也有右偏置结构right dislocation。

下面简单分析一下这个：

句子的意思是：我们一开始看到的是，去年开始的凶案增加的趋势还没有消失。

that that homicide increase that began last year，第1个that是连词，第2个是指示代词（限定词）“那个”，第3个是关系代词。

Work is where people learn to manage social interactions, not define them out of existence.

这个句子not并列了两个谓语部分：

Work is where people (learn to manage social interactions, not define them out of existence).

这属于一种无连词并列。

比如，

He is your friend, not your slave.

这种并列是不是分句层面上的，而是次分句级别(subclausal)，不需要补全。

【一】What we are beginning to see is that that homicide increases that began last year, it has not disappeared.

解答如下:

【1】第1个that引导表语从句

【2】第2个that为修饰限定词，修饰限定后面名词“homicide”

【3】第3个that为关系代词引导定语从句

备注：what为融合性连接代词引导主语从句，并在主语从句作to see的宾语成分。

【二】work is where people learn to manage social interactions, not define them out of existence.

参考译文：工作是人们学习完成社交互动的地方，而不是人们学着把社交互动定义为不存在的地方。

解答：work is where people learn to manage social interactions and it is not where people learn to define them out of existence.