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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-3 - 2q-2 + 7q-1 - 10 + 17q - 20q2 + 21q3 - 19q4 + 15q5 - 10q6 + 5q7 - q8

A2 (sl(3)) Invariant:

q-10 + q-8 + q-6 + 6q-4 + 3q-2 + 4 + 8q2 - q4 + 4q6 - q8 - q10 + 2q12 - 4q14 + 4q16 - q18 - q20 + 3q22 - q24

HOMFLY-PT Polynomial:

a-6 - a-6z4 - a-4 - a-4z2 + a-4z4 + a-4z6 + a-2z-2 + a-2 + a-2z4 + a-2z6 - 2z-2 - 3 - 4z2 - 2z4 + a2z-2 + 2a2 + a2z2

Kauffman Polynomial:

a-9z5 - 5a-8z4 + 5a-8z6 + 4a-7z3 - 15a-7z5 + 10a-7z7 - 2a-6 + 6a-6z2 - 4a-6z4 - 11a-6z6 + 10a-6z8 - 4a-5z + 18a-5z3 - 28a-5z5 + 8a-5z7 + 5a-5z9 - 2a-4 + 12a-4z2 - a-4z4 - 24a-4z6 + 15a-4z8 + a-4z10 - 12a-3z + 34a-3z3 - 30a-3z5 + 2a-3z7 + 7a-3z9 + a-2z-2 - 2a-2 + 6a-2z2 - 13a-2z6 + 8a-2z8 + a-2z10 - 2a-1z-1 - 4a-1z + 20a-1z3 - 22a-1z5 + 6a-1z7 + 2a-1z9 + 2z-2 - 5 + 6z2 - 2z4 - 4z6 + 3z8 - 2az-1 + 4az - 4az5 + 2az7 + a2z-2 - 4a2 + 6a2z2 - 4a2z4 + a2z6

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 = 17                       1 j = 15                     4   j = 13                   6 1   j = 11                 9 4     j = 9               11 7       j = 7             10 8         j = 5           10 11           j = 3         7 10             j = 1       5 12               j = -1     2 5                 j = -3     5                   j = -5 1 2                     j = -7 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, 481]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 481]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[14, 8, 15, 7], X[20, 16, 21, 15], > X[18, 10, 19, 9], X[8, 18, 9, 17], X[22, 20, 17, 19], X[16, 22, 5, 21], > X[10, 14, 11, 13], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {6, -5, 7, -4, 8, -7}, > {10, -1, 3, -6, 5, -9, 11, -2, 9, -3, 4, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-3 2 7 2 3 4 5 6 7 8 -10 + q - -- + - + 17 q - 20 q + 21 q - 19 q + 15 q - 10 q + 5 q - q 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 -8 -6 6 3 2 4 6 8 10 12 14 4 + q + q + q + -- + -- + 8 q - q + 4 q - q - q + 2 q - 4 q + 4 2 q q 16 18 20 22 24 > 4 q - q - q + 3 q - q
In[8]:=	HOMFLYPT[Link[11, Alternating, 481]][a, z]
Out[8]=	2 2 4 -6 -4 -2 2 2 1 a 2 z 2 2 4 z -3 + a - a + a + 2 a - -- + ----- + -- - 4 z - -- + a z - 2 z - -- + 2 2 2 2 4 6 z a z z a a 4 4 6 6 z z z z > -- + -- + -- + -- 4 2 4 2 a a a a
In[9]:=	Kauffman[Link[11, Alternating, 481]][a, z]
Out[9]=	2 2 2 2 2 2 1 a 2 2 a 4 z 12 z 4 z -5 - -- - -- - -- - 4 a + -- + ----- + -- - --- - --- - --- - ---- - --- + 6 4 2 2 2 2 2 a z z 5 3 a a a a z a z z a a 2 2 2 3 3 3 2 6 z 12 z 6 z 2 2 4 z 18 z 34 z > 4 a z + 6 z + ---- + ----- + ---- + 6 a z + ---- + ----- + ----- + 6 4 2 7 5 3 a a a a a a 3 4 4 4 5 5 5 5 20 z 4 5 z 4 z z 2 4 z 15 z 28 z 30 z > ----- - 2 z - ---- - ---- - -- - 4 a z + -- - ----- - ----- - ----- - a 8 6 4 9 7 5 3 a a a a a a a 5 6 6 6 6 7 22 z 5 6 5 z 11 z 24 z 13 z 2 6 10 z > ----- - 4 a z - 4 z + ---- - ----- - ----- - ----- + a z + ----- + a 8 6 4 2 7 a a a a a 7 7 7 8 8 8 9 9 8 z 2 z 6 z 7 8 10 z 15 z 8 z 5 z 7 z > ---- + ---- + ---- + 2 a z + 3 z + ----- + ----- + ---- + ---- + ---- + 5 3 a 6 4 2 5 3 a a a a a a a 9 10 10 2 z z z > ---- + --- + --- a 4 2 a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 2 5 2 5 5 q 3 12 q + 7 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 10 q t + 7 4 5 4 5 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 10 q t + 11 q t + 10 q t + 8 q t + 11 q t + 7 q t + 9 q t + 11 5 13 5 13 6 15 6 17 7 > 4 q t + 6 q t + q t + 4 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a481

L11a480 L11a482