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L11n371 L11n373

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DrawMorseLink

PD Presentation:

X6172 X3,10,4,11 X7,19,8,18 X15,17,16,22 X9,20,10,21 X13,9,14,8 X17,15,18,14 X21,5,22,16 X19,12,20,13 X2536 X11,4,12,1

Gauss Code:

{{1, -10, -2, 11}, {-7, 3, -9, 5, -8, 4}, {10, -1, -3, 6, -5, 2, -11, 9, -6, 7, -4, 8}}

Jones Polynomial:

2q-5 - 4q-4 + 7q-3 - 8q-2 + 10q-1 - 8 + 8q - 5q2 + 3q3 - q4

A2 (sl(3)) Invariant:

q-20 + q-16 + 4q-10 + 2q-8 + 6q-6 + 4q-4 + 3q-2 + 4 + 2q4 + q10 - q12

HOMFLY-PT Polynomial:

- 2a-2z2 - a-2z4 + z-2 + 2 + 2z2 + 3z4 + z6 - 2a2z-2 - 2a2 + a2z2 + 3a2z4 + a2z6 + a4z-2 - a4 - 3a4z2 - a4z4 + a6

Kauffman Polynomial:

- 2a-3z + 5a-3z3 - 4a-3z5 + a-3z7 + 2a-2 - 8a-2z2 + 17a-2z4 - 13a-2z6 + 3a-2z8 - 7a-1z + 21a-1z3 - 12a-1z5 - 3a-1z7 + 2a-1z9 - z-2 + 10 - 28z2 + 45z4 - 36z6 + 9z8 + 2az-1 - 14az + 30az3 - 25az5 + 2az7 + 2az9 - 2a2z-2 + 12a2 - 26a2z2 + 27a2z4 - 20a2z6 + 6a2z8 + 2a3z-1 - 10a3z + 17a3z3 - 16a3z5 + 6a3z7 - a4z-2 + 4a4 - 3a4z2 - a4z4 + 3a4z6 - a5z + 3a5z3 + a5z5 - a6 + 3a6z2

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 9                   1 j = 7                 2   j = 5               3 1   j = 3             5 2     j = 1           4 4       j = -1         6 4         j = -3       3 5           j = -5     4 5             j = -7   1 4               j = -9 1 3                 j = -11 2

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Length[Skeleton[L]]
Out[2]=	3
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 372]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 372]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 10, 4, 11], X[7, 19, 8, 18], X[15, 17, 16, 22], > X[9, 20, 10, 21], X[13, 9, 14, 8], X[17, 15, 18, 14], X[21, 5, 22, 16], > X[19, 12, 20, 13], X[2, 5, 3, 6], X[11, 4, 12, 1]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -2, 11}, {-7, 3, -9, 5, -8, 4}, > {10, -1, -3, 6, -5, 2, -11, 9, -6, 7, -4, 8}]
In[6]:=	Jones[L][q]
Out[6]=	2 4 7 8 10 2 3 4 -8 + -- - -- + -- - -- + -- + 8 q - 5 q + 3 q - q 5 4 3 2 q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-20 -16 4 2 6 4 3 4 10 12 4 + q + q + --- + -- + -- + -- + -- + 2 q + q - q 10 8 6 4 2 q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 372]][a, z]
Out[8]=	2 4 2 2 4 6 -2 2 a a 2 2 z 2 2 4 2 4 2 - 2 a - a + a + z - ---- + -- + 2 z - ---- + a z - 3 a z + 3 z - 2 2 2 z z a 4 z 2 4 4 4 6 2 6 > -- + 3 a z - a z + z + a z 2 a
In[9]:=	Kauffman[Link[11, NonAlternating, 372]][a, z]
Out[9]=	2 4 3 2 2 4 6 -2 2 a a 2 a 2 a 2 z 7 z 10 + -- + 12 a + 4 a - a - z - ---- - -- + --- + ---- - --- - --- - 2 2 2 z z 3 a a z z a 2 3 5 2 8 z 2 2 4 2 6 2 > 14 a z - 10 a z - a z - 28 z - ---- - 26 a z - 3 a z + 3 a z + 2 a 3 3 4 5 z 21 z 3 3 3 5 3 4 17 z 2 4 > ---- + ----- + 30 a z + 17 a z + 3 a z + 45 z + ----- + 27 a z - 3 a 2 a a 5 5 6 4 4 4 z 12 z 5 3 5 5 5 6 13 z > a z - ---- - ----- - 25 a z - 16 a z + a z - 36 z - ----- - 3 a 2 a a 7 7 8 2 6 4 6 z 3 z 7 3 7 8 3 z 2 8 > 20 a z + 3 a z + -- - ---- + 2 a z + 6 a z + 9 z + ---- + 6 a z + 3 a 2 a a 9 2 z 9 > ---- + 2 a z a
In[10]:=	Kh[L][q, t]
Out[10]=	5 6 2 1 3 1 4 4 5 3 4 t -- + - + ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + --- + 3 q 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q q q t q t q t q t q t q t q t q t 2 3 2 3 3 5 3 5 4 7 4 9 5 > 4 q t + 4 q t + 5 q t + 2 q t + 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n372

L11n371 L11n373