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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-2 + 5q-1 - 11 + 18q - 22q2 + 27q3 - 25q4 + 22q5 - 15q6 + 9q7 - 4q8 + q9

A2 (sl(3)) Invariant:

- q-6 + 3q-4 - q-2 - 2 + 5q2 - 4q4 + 5q6 + 4q8 + 3q10 + 9q12 + 7q16 - 3q20 + 3q22 - 2q24 + q28

HOMFLY-PT Polynomial:

a-8 + a-8z2 + a-6z-2 - 3a-6 - 6a-6z2 - 3a-6z4 - 2a-4z-2 + 2a-4 + 8a-4z2 + 6a-4z4 + 2a-4z6 + a-2z-2 - a-2 - 3a-2z2 + a-2z6 + 1 - z4

Kauffman Polynomial:

a-10z2 - 2a-10z4 + a-10z6 - 2a-9z + 6a-9z3 - 9a-9z5 + 4a-9z7 + 2a-8 - 6a-8z2 + 11a-8z4 - 15a-8z6 + 7a-8z8 - 7a-7z + 23a-7z3 - 20a-7z5 - 2a-7z7 + 6a-7z9 + a-6z-2 + 3a-6 - 25a-6z2 + 60a-6z4 - 59a-6z6 + 19a-6z8 + 2a-6z10 - 2a-5z-1 - 5a-5z + 29a-5z3 - 18a-5z5 - 17a-5z7 + 15a-5z9 + 2a-4z-2 + 2a-4 - 27a-4z2 + 65a-4z4 - 70a-4z6 + 26a-4z8 + 2a-4z10 - 2a-3z-1 - a-3z + 15a-3z3 - 22a-3z5 + 9a-3z9 + a-2z-2 + a-2 - 9a-2z2 + 14a-2z4 - 22a-2z6 + 14a-2z8 - a-1z + 3a-1z3 - 14a-1z5 + 11a-1z7 + 1 - 4z4 + 5z6 + az5

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 j = 19                       1 j = 17                     3   j = 15                   6 1   j = 13                 9 3     j = 11               13 6       j = 9             14 11         j = 7           13 11           j = 5         9 14             j = 3       9 13               j = 1     4 11                 j = -1   1 7                   j = -3   4                     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, 483]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 483]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 7, 15, 8], X[4, 15, 1, 16], X[10, 6, 11, 5], > X[8, 4, 9, 3], X[22, 18, 19, 17], X[20, 12, 21, 11], X[12, 20, 13, 19], > X[18, 22, 5, 21], X[16, 10, 17, 9], X[2, 14, 3, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -3}, {8, -7, 9, -6}, > {4, -1, 2, -5, 10, -4, 7, -8, 11, -2, 3, -10, 6, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-2 5 2 3 4 5 6 7 8 9 -11 - q + - + 18 q - 22 q + 27 q - 25 q + 22 q - 15 q + 9 q - 4 q + q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-6 3 -2 2 4 6 8 10 12 16 -2 - q + -- - q + 5 q - 4 q + 5 q + 4 q + 3 q + 9 q + 7 q - 4 q 20 22 24 28 > 3 q + 3 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 483]][a, z]
Out[8]=	2 2 2 2 -8 3 2 -2 1 2 1 z 6 z 8 z 3 z 1 + a - -- + -- - a + ----- - ----- + ----- + -- - ---- + ---- - ---- - 6 4 6 2 4 2 2 2 8 6 4 2 a a a z a z a z a a a a 4 4 6 6 4 3 z 6 z 2 z z > z - ---- + ---- + ---- + -- 6 4 4 2 a a a a
In[9]:=	Kauffman[Link[11, Alternating, 483]][a, z]
Out[9]=	2 3 2 -2 1 2 1 2 2 2 z 7 z 1 + -- + -- + -- + a + ----- + ----- + ----- - ---- - ---- - --- - --- - 8 6 4 6 2 4 2 2 2 5 3 9 7 a a a a z a z a z a z a z a a 2 2 2 2 2 3 3 3 5 z z z z 6 z 25 z 27 z 9 z 6 z 23 z 29 z > --- - -- - - + --- - ---- - ----- - ----- - ---- + ---- + ----- + ----- + 5 3 a 10 8 6 4 2 9 7 5 a a a a a a a a a a 3 3 4 4 4 4 4 5 5 15 z 3 z 4 2 z 11 z 60 z 65 z 14 z 9 z 20 z > ----- + ---- - 4 z - ---- + ----- + ----- + ----- + ----- - ---- - ----- - 3 a 10 8 6 4 2 9 7 a a a a a a a a 5 5 5 6 6 6 6 6 18 z 22 z 14 z 5 6 z 15 z 59 z 70 z 22 z > ----- - ----- - ----- + a z + 5 z + --- - ----- - ----- - ----- - ----- + 5 3 a 10 8 6 4 2 a a a a a a a 7 7 7 7 8 8 8 8 9 9 4 z 2 z 17 z 11 z 7 z 19 z 26 z 14 z 6 z 15 z > ---- - ---- - ----- + ----- + ---- + ----- + ----- + ----- + ---- + ----- + 9 7 5 a 8 6 4 2 7 5 a a a a a a a a a 9 10 10 9 z 2 z 2 z > ---- + ----- + ----- 3 6 4 a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 4 1 7 4 q 3 5 5 2 11 q + 9 q + ----- + ----- + ---- + --- + --- + 13 q t + 9 q t + 14 q t + 5 3 3 2 2 q t t q t q t q t 7 2 7 3 9 3 9 4 11 4 11 5 > 13 q t + 11 q t + 14 q t + 11 q t + 13 q t + 6 q t + 13 5 13 6 15 6 15 7 17 7 19 8 > 9 q t + 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a483

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