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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-5 + 3q-4 - 6q-3 + 10q-2 - 10q-1 + 12 - 9q + 8q2 - 4q3 + q4

A2 (sl(3)) Invariant:

- q-16 + q-12 - 2q-10 + 3q-8 + 2q-6 + 3q-4 + 7q-2 + 3 + 7q2 + q4 + 2q6 + 2q8 - 2q10 + q12

HOMFLY-PT Polynomial:

a-2z-2 + a-2 + a-2z2 + a-2z4 - 2z-2 - 3 - 4z2 - 3z4 - z6 + a2z-2 + 3a2 + 4a2z2 + 2a2z4 - a4 - a4z2

Kauffman Polynomial:

a-4z4 - 2a-3z3 + 4a-3z5 + a-2z-2 - 2a-2 + 5a-2z2 - 11a-2z4 + 8a-2z6 - 2a-1z-1 + a-1z - 7a-1z5 + 7a-1z7 + 2z-2 - 5 + 17z2 - 28z4 + 12z6 + 2z8 - 2az-1 + 3az + 3az3 - 17az5 + 11az7 + a2z-2 - 6a2 + 17a2z2 - 22a2z4 + 7a2z6 + 2a2z8 + 3a3z - a3z3 - 5a3z5 + 4a3z7 - 2a4 + 5a4z2 - 6a4z4 + 3a4z6 + a5z - 2a5z3 + a5z5

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 9                   1 j = 7                 3   j = 5               5 1   j = 3             4 3     j = 1           8 5       j = -1         6 8         j = -3       4 4           j = -5     2 6             j = -7   1 4               j = -9   2                 j = -11 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[9, Alternating, 53]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[9, Alternating, 53]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 7, 13, 8], X[4, 13, 1, 14], X[18, 10, 15, 9], > X[8, 4, 9, 3], X[16, 5, 17, 6], X[14, 17, 5, 18], X[10, 16, 11, 15], > X[2, 12, 3, 11]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 5, -3}, {8, -6, 7, -4}, {6, -1, 2, -5, 4, -8, 9, -2, 3, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-5 3 6 10 10 2 3 4 12 - q + -- - -- + -- - -- - 9 q + 8 q - 4 q + q 4 3 2 q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-16 -12 2 3 2 3 7 2 4 6 8 10 3 - q + q - --- + -- + -- + -- + -- + 7 q + q + 2 q + 2 q - 2 q + 10 8 6 4 2 q q q q q 12 > q
In[8]:=	HOMFLYPT[Link[9, Alternating, 53]][a, z]
Out[8]=	2 2 -2 2 4 2 1 a 2 z 2 2 4 2 4 -3 + a + 3 a - a - -- + ----- + -- - 4 z + -- + 4 a z - a z - 3 z + 2 2 2 2 2 z a z z a 4 z 2 4 6 > -- + 2 a z - z 2 a
In[9]:=	Kauffman[Link[9, Alternating, 53]][a, z]
Out[9]=	2 2 2 4 2 1 a 2 2 a z 3 -5 - -- - 6 a - 2 a + -- + ----- + -- - --- - --- + - + 3 a z + 3 a z + 2 2 2 2 2 a z z a a z a z z 2 3 5 2 5 z 2 2 4 2 2 z 3 3 3 > a z + 17 z + ---- + 17 a z + 5 a z - ---- + 3 a z - a z - 2 3 a a 4 4 5 5 5 3 4 z 11 z 2 4 4 4 4 z 7 z 5 > 2 a z - 28 z + -- - ----- - 22 a z - 6 a z + ---- - ---- - 17 a z - 4 2 3 a a a a 6 7 3 5 5 5 6 8 z 2 6 4 6 7 z 7 > 5 a z + a z + 12 z + ---- + 7 a z + 3 a z + ---- + 11 a z + 2 a a 3 7 8 2 8 > 4 a z + 2 z + 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	8 1 2 1 4 2 6 4 4 6 - + 8 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 9 4 > 5 q t + 4 q t + 3 q t + 5 q t + q t + 3 q t + q t

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