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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X22,12,9,11 X20,14,21,13 X18,16,19,15 X8,18,5,17 X16,8,17,7 X14,20,15,19 X12,22,13,21 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

q-4 - q-3 + 4q-2 - 4q-1 + 7 - 7q + 8q2 - 7q3 + 5q4 - 4q5 + 3q6 - q7

A2 (sl(3)) Invariant:

q-14 + 2q-12 + 2q-10 + 5q-8 + 5q-6 + 3q-4 + 5q-2 + 2 + q2 + q4 + q8 - q10 - q12 + q14 - q16 + q18 + q20 - q22

HOMFLY-PT Polynomial:

- a-6z2 + a-4z2 + a-4z4 + a-2z4 + z-2 + 2 + z2 + z4 - 2a2z-2 - 3a2 - 2a2z2 + a4z-2 + a4

Kauffman Polynomial:

3a-7z3 - 4a-7z5 + a-7z7 - 3a-6z2 + 16a-6z4 - 14a-6z6 + 3a-6z8 - 7a-5z3 + 18a-5z5 - 14a-5z7 + 3a-5z9 - 2a-4z2 + 10a-4z4 - 8a-4z6 - a-4z8 + a-4z10 - 10a-3z3 + 22a-3z5 - 17a-3z7 + 4a-3z9 + a-2z2 - 6a-2z4 + 5a-2z6 - 3a-2z8 + a-2z10 - a-1z7 + a-1z9 - z-2 + 3 - 3z2 + z4 + z8 + 2az-1 - 3az + az5 + az7 - 2a2z-2 + 5a2 - 6a2z2 + 2a2z4 + a2z6 + 2a3z-1 - 3a3z + a3z5 - a4z-2 + 3a4 - 3a4z2 + a4z4

Khovanov Homology:

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 15                       1 j = 13                     2   j = 11                   2 1   j = 9                 3 2     j = 7               4 2       j = 5             4 3         j = 3           3 4           j = 1         4 4             j = -1       3 6               j = -3     1 1                 j = -5     3                   j = -7 1 1                     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, 397]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 397]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[22, 12, 9, 11], X[20, 14, 21, 13], > X[18, 16, 19, 15], X[8, 18, 5, 17], X[16, 8, 17, 7], X[14, 20, 15, 19], > X[12, 22, 13, 21], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 7, -6}, > {11, -2, 3, -9, 4, -8, 5, -7, 6, -5, 8, -4, 9, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-4 -3 4 4 2 3 4 5 6 7 7 + q - q + -- - - - 7 q + 8 q - 7 q + 5 q - 4 q + 3 q - q 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-14 2 2 5 5 3 5 2 4 8 10 12 14 2 + q + --- + --- + -- + -- + -- + -- + q + q + q - q - q + q - 12 10 8 6 4 2 q q q q q q 16 18 20 22 > q + q + q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 397]][a, z]
Out[8]=	2 4 2 2 4 4 2 4 -2 2 a a 2 z z 2 2 4 z z 2 - 3 a + a + z - ---- + -- + z - -- + -- - 2 a z + z + -- + -- 2 2 6 4 4 2 z z a a a a
In[9]:=	Kauffman[Link[11, Alternating, 397]][a, z]
Out[9]=	2 4 3 2 2 4 -2 2 a a 2 a 2 a 3 2 3 z 3 + 5 a + 3 a - z - ---- - -- + --- + ---- - 3 a z - 3 a z - 3 z - ---- - 2 2 z z 6 z z a 2 2 3 3 3 4 4 2 z z 2 2 4 2 3 z 7 z 10 z 4 16 z 10 z > ---- + -- - 6 a z - 3 a z + ---- - ---- - ----- + z + ----- + ----- - 4 2 7 5 3 6 4 a a a a a a a 4 5 5 5 6 6 z 2 4 4 4 4 z 18 z 22 z 5 3 5 14 z > ---- + 2 a z + a z - ---- + ----- + ----- + a z + a z - ----- - 2 7 5 3 6 a a a a a 6 6 7 7 7 7 8 8 8 z 5 z 2 6 z 14 z 17 z z 7 8 3 z z > ---- + ---- + a z + -- - ----- - ----- - -- + a z + z + ---- - -- - 4 2 7 5 3 a 6 4 a a a a a a a 8 9 9 9 10 10 3 z 3 z 4 z z z z > ---- + ---- + ---- + -- + --- + --- 2 5 3 a 4 2 a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	6 1 1 1 3 1 1 3 3 - + 4 q + ----- + ----- + ----- + ----- + ----- + ---- + --- + 4 q t + 3 q t + q 9 4 7 4 7 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 9 5 > 4 q t + 4 q t + 3 q t + 4 q t + 2 q t + 3 q t + 2 q t + 11 5 11 6 13 6 15 7 > 2 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a397

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