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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-2 - q-1 + 4 - 4q + 6q2 - 7q3 + 8q4 - 7q5 + 6q6 - 4q7 + 3q8 - q9

A2 (sl(3)) Invariant:

q-6 + 2q-4 + 2q-2 + 4 + 3q2 + 3q4 + 2q6 + q8 + 3q10 + 3q14 + q16 + q18 + q20 + q24 - q26

HOMFLY-PT Polynomial:

- 3a-6z2 - 4a-6z4 - a-6z6 + a-4z-2 + 5a-4 + 11a-4z2 + 12a-4z4 + 6a-4z6 + a-4z8 - 2a-2z-2 - 11a-2 - 18a-2z2 - 11a-2z4 - 2a-2z6 + z-2 + 6 + 5z2 + z4

Kauffman Polynomial:

a-11z3 - 2a-10z2 + 3a-10z4 - 3a-9z3 + 4a-9z5 + a-8 - 2a-8z2 - 3a-8z4 + 4a-8z6 - 6a-7z5 + 4a-7z7 - 3a-6z2 + 8a-6z4 - 11a-6z6 + 4a-6z8 - 2a-5z3 + 8a-5z5 - 10a-5z7 + 3a-5z9 - a-4z-2 + 5a-4 - 10a-4z2 + 19a-4z4 - 13a-4z6 + a-4z10 + 2a-3z-1 - 11a-3z + 6a-3z3 + 18a-3z5 - 18a-3z7 + 4a-3z9 - 2a-2z-2 + 13a-2 - 25a-2z2 + 22a-2z4 - 5a-2z6 - 3a-2z8 + a-2z10 + 2a-1z-1 - 11a-1z + 12a-1z3 - 4a-1z7 + a-1z9 - z-2 + 8 - 18z2 + 17z4 - 7z6 + z8

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 = 19                       1 j = 17                     2   j = 15                   2 1   j = 13                 4 2     j = 11               5 4       j = 9             3 2         j = 7           4 5           j = 5         2 3             j = 3       3 5               j = 1     1 1                 j = -1     3                   j = -3 1 1                     j = -5 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, 442]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 442]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 3, 15, 4], X[22, 16, 13, 15], X[8, 18, 9, 17], > X[16, 8, 17, 7], X[18, 10, 19, 9], X[20, 12, 21, 11], X[10, 20, 11, 19], > X[12, 22, 5, 21], X[2, 5, 3, 6], X[4, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 5, -4, 6, -8, 7, -9}, > {11, -2, 3, -5, 4, -6, 8, -7, 9, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-2 1 2 3 4 5 6 7 8 9 4 + q - - - 4 q + 6 q - 7 q + 8 q - 7 q + 6 q - 4 q + 3 q - q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 2 2 2 4 6 8 10 14 16 18 20 4 + q + -- + -- + 3 q + 3 q + 2 q + q + 3 q + 3 q + q + q + q + 4 2 q q 24 26 > q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 442]][a, z]
Out[8]=	2 2 2 4 5 11 -2 1 2 2 3 z 11 z 18 z 4 4 z 6 + -- - -- + z + ----- - ----- + 5 z - ---- + ----- - ----- + z - ---- + 4 2 4 2 2 2 6 4 2 6 a a a z a z a a a a 4 4 6 6 6 8 12 z 11 z z 6 z 2 z z > ----- - ----- - -- + ---- - ---- + -- 4 2 6 4 2 4 a a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 442]][a, z]
Out[9]=	-8 5 13 -2 1 2 2 2 11 z 11 z 2 8 + a + -- + -- - z - ----- - ----- + ---- + --- - ---- - ---- - 18 z - 4 2 4 2 2 2 3 a z 3 a a a a z a z a z a 2 2 2 2 2 3 3 3 3 3 2 z 2 z 3 z 10 z 25 z z 3 z 2 z 6 z 12 z > ---- - ---- - ---- - ----- - ----- + --- - ---- - ---- + ---- + ----- + 10 8 6 4 2 11 9 5 3 a a a a a a a a a a 4 4 4 4 4 5 5 5 5 4 3 z 3 z 8 z 19 z 22 z 4 z 6 z 8 z 18 z > 17 z + ---- - ---- + ---- + ----- + ----- + ---- - ---- + ---- + ----- - 10 8 6 4 2 9 7 5 3 a a a a a a a a a 6 6 6 6 7 7 7 7 6 4 z 11 z 13 z 5 z 4 z 10 z 18 z 4 z 8 > 7 z + ---- - ----- - ----- - ---- + ---- - ----- - ----- - ---- + z + 8 6 4 2 7 5 3 a a a a a a a a 8 8 9 9 9 10 10 4 z 3 z 3 z 4 z z z z > ---- - ---- + ---- + ---- + -- + --- + --- 6 2 5 3 a 4 2 a a a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 3 5 1 1 1 3 q q 3 q 5 7 5 q + 2 q + ----- + ----- + ----- + ---- + -- + - + ---- + 3 q t + 4 q t + 5 4 3 4 3 3 2 2 t t q t q t q t q t t 7 2 9 2 9 3 11 3 11 4 13 4 13 5 > 5 q t + 3 q t + 2 q t + 5 q t + 4 q t + 4 q t + 2 q t + 15 5 15 6 17 6 19 7 > 2 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a442

L11a441 L11a443