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DrawMorseLink

PD Presentation:

X6172 X14,6,15,5 X8493 X2,16,3,15 X16,7,17,8 X18,10,19,9 X20,11,21,12 X22,20,13,19 X12,14,5,13 X4,17,1,18 X10,21,11,22

Gauss Code:

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

Jones Polynomial:

- q-4 + 4q-3 - 7q-2 + 12q-1 - 14 + 18q - 17q2 + 16q3 - 11q4 + 7q5 - 4q6 + q7

A2 (sl(3)) Invariant:

- q-12 + q-10 + 2q-8 + q-6 + 5q-4 + q-2 + 5 + 5q2 + 2q4 + 7q6 - q8 + 3q10 - q12 - 2q14 + q16 - 2q18 + q20

HOMFLY-PT Polynomial:

- a-4 + a-4z2 + 3a-4z4 + a-4z6 + a-2z-2 + 3a-2 - 3a-2z2 - 8a-2z4 - 5a-2z6 - a-2z8 - 2z-2 - 3 + 4z2 + 7z4 + 2z6 + a2z-2 + a2 - 2a2z2 - a2z4

Kauffman Polynomial:

a-8z4 - 3a-7z3 + 4a-7z5 + 2a-6z2 - 7a-6z4 + 7a-6z6 + a-5z - 3a-5z3 - 6a-5z5 + 8a-5z7 - 2a-4 + 5a-4z2 - 4a-4z4 - 8a-4z6 + 8a-4z8 + 3a-3z + 2a-3z3 - 5a-3z5 - 6a-3z7 + 6a-3z9 + a-2z-2 - 6a-2 + a-2z2 + 28a-2z4 - 36a-2z6 + 9a-2z8 + 2a-2z10 - 2a-1z-1 + 3a-1z + 19a-1z5 - 31a-1z7 + 11a-1z9 + 2z-2 - 5 - 7z2 + 40z4 - 36z6 + 5z8 + 2z10 - 2az-1 + az + 11az5 - 16az7 + 5az9 + a2z-2 - 2a2 - 5a2z2 + 16a2z4 - 15a2z6 + 4a2z8 + 2a3z3 - 3a3z5 + a3z7

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 15                       1 j = 13                     3   j = 11                   4 1   j = 9                 7 3     j = 7               9 4       j = 5             8 7         j = 3           10 9           j = 1         8 12             j = -1       4 6               j = -3     3 8                 j = -5   1 4                   j = -7   3                     j = -9 1

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, Alternating, 431]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 431]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 6, 15, 5], X[8, 4, 9, 3], X[2, 16, 3, 15], > X[16, 7, 17, 8], X[18, 10, 19, 9], X[20, 11, 21, 12], X[22, 20, 13, 19], > X[12, 14, 5, 13], X[4, 17, 1, 18], X[10, 21, 11, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -4, 3, -10}, {2, -1, 5, -3, 6, -11, 7, -9}, > {9, -2, 4, -5, 10, -6, 8, -7, 11, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-4 4 7 12 2 3 4 5 6 7 -14 - q + -- - -- + -- + 18 q - 17 q + 16 q - 11 q + 7 q - 4 q + q 3 2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -10 2 -6 5 -2 2 4 6 8 10 12 5 - q + q + -- + q + -- + q + 5 q + 2 q + 7 q - q + 3 q - q - 8 4 q q 14 16 18 20 > 2 q + q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 431]][a, z]
Out[8]=	2 2 2 -4 3 2 2 1 a 2 z 3 z 2 2 4 -3 - a + -- + a - -- + ----- + -- + 4 z + -- - ---- - 2 a z + 7 z + 2 2 2 2 2 4 2 a z a z z a a 4 4 6 6 8 3 z 8 z 2 4 6 z 5 z z > ---- - ---- - a z + 2 z + -- - ---- - -- 4 2 4 2 2 a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 431]][a, z]
Out[9]=	2 2 6 2 2 1 a 2 2 a z 3 z 3 z -5 - -- - -- - 2 a + -- + ----- + -- - --- - --- + -- + --- + --- + a z - 4 2 2 2 2 2 a z z 5 3 a a a z a z z a a 2 2 2 3 3 3 2 2 z 5 z z 2 2 3 z 3 z 2 z 3 3 4 > 7 z + ---- + ---- + -- - 5 a z - ---- - ---- + ---- + 2 a z + 40 z + 6 4 2 7 5 3 a a a a a a 4 4 4 4 5 5 5 5 z 7 z 4 z 28 z 2 4 4 z 6 z 5 z 19 z > -- - ---- - ---- + ----- + 16 a z + ---- - ---- - ---- + ----- + 8 6 4 2 7 5 3 a a a a a a a a 6 6 6 7 7 5 3 5 6 7 z 8 z 36 z 2 6 8 z 6 z > 11 a z - 3 a z - 36 z + ---- - ---- - ----- - 15 a z + ---- - ---- - 6 4 2 5 3 a a a a a 7 8 8 9 9 31 z 7 3 7 8 8 z 9 z 2 8 6 z 11 z > ----- - 16 a z + a z + 5 z + ---- + ---- + 4 a z + ---- + ----- + a 4 2 3 a a a a 10 9 10 2 z > 5 a z + 2 z + ----- 2 a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 3 1 4 3 8 4 6 12 q + 10 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t q t q t q t q t q t q t q t 8 q 3 5 5 2 7 2 7 3 9 3 9 4 > --- + 9 q t + 8 q t + 7 q t + 9 q t + 4 q t + 7 q t + 3 q t + t 11 4 11 5 13 5 15 6 > 4 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a431

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