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Knotscape

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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X15,20,16,21 X14,8,15,7 X10,22,5,21 X18,11,19,12 X9,17,10,16 X22,17,11,18 X19,9,20,8 X2536 X4,14,1,13

Gauss Code:

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

Jones Polynomial:

- q-3 + 2q-2 - 4q-1 + 6 - 5q + 7q2 - 4q3 + 4q4 - 2q5 + q6

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

2a-4z-2 + 3a-4 + 3a-4z2 + a-4z4 - 5a-2z-2 - 10a-2 - 10a-2z2 - 5a-2z4 - a-2z6 + 4z-2 + 9 + 7z2 + 2z4 - a2z-2 - 2a2 - a2z2

Kauffman Polynomial:

3a-6z2 - 4a-6z4 + a-6z6 + 4a-5z3 - 7a-5z5 + 2a-5z7 - 2a-4z-2 + 10a-4 - 22a-4z2 + 23a-4z4 - 14a-4z6 + 3a-4z8 + 5a-3z-1 - 16a-3z + 23a-3z3 - 13a-3z5 + a-3z9 - 5a-2z-2 + 20a-2 - 44a-2z2 + 48a-2z4 - 25a-2z6 + 5a-2z8 + 9a-1z-1 - 27a-1z + 30a-1z3 - 10a-1z5 - a-1z7 + a-1z9 - 4z-2 + 13 - 22z2 + 23z4 - 10z6 + 2z8 + 5az-1 - 13az + 12az3 - 4az5 + az7 - a2z-2 + 2a2 - 3a2z2 + 2a2z4 + a3z-1 - 2a3z + a3z3

Khovanov Homology:

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                   1 j = 11                 2 1 j = 9               2     j = 7             2 2     j = 5           5 2       j = 3         1 3         j = 1       5 4           j = -1     1 3             j = -3   1 3               j = -5   1                 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, NonAlternating, 296]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 296]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[15, 20, 16, 21], X[14, 8, 15, 7], > X[10, 22, 5, 21], X[18, 11, 19, 12], X[9, 17, 10, 16], X[22, 17, 11, 18], > X[19, 9, 20, 8], X[2, 5, 3, 6], X[4, 14, 1, 13]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, 9, -7, -5}, > {6, -2, 11, -4, -3, 7, 8, -6, -9, 3, 5, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-3 2 4 2 3 4 5 6 6 - q + -- - - - 5 q + 7 q - 4 q + 4 q - 2 q + q 2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 -8 -6 3 2 4 6 8 10 12 14 18 -q - q - q - -- + 3 q + 6 q + 6 q + 8 q + 4 q + 3 q + 2 q + q 4 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 296]][a, z]
Out[8]=	2 2 2 3 10 2 4 2 5 a 2 3 z 10 z 2 2 9 + -- - -- - 2 a + -- + ----- - ----- - -- + 7 z + ---- - ----- - a z + 4 2 2 4 2 2 2 2 4 2 a a z a z a z z a a 4 4 6 4 z 5 z z > 2 z + -- - ---- - -- 4 2 2 a a a
In[9]:=	Kauffman[Link[11, NonAlternating, 296]][a, z]
Out[9]=	2 3 10 20 2 4 2 5 a 5 9 5 a a 16 z 13 + -- + -- + 2 a - -- - ----- - ----- - -- + ---- + --- + --- + -- - ---- - 4 2 2 4 2 2 2 2 3 a z z z 3 a a z a z a z z a z a 2 2 2 3 27 z 3 2 3 z 22 z 44 z 2 2 4 z > ---- - 13 a z - 2 a z - 22 z + ---- - ----- - ----- - 3 a z + ---- + a 6 4 2 5 a a a a 3 3 4 4 4 23 z 30 z 3 3 3 4 4 z 23 z 48 z 2 4 > ----- + ----- + 12 a z + a z + 23 z - ---- + ----- + ----- + 2 a z - 3 a 6 4 2 a a a a 5 5 5 6 6 6 7 7 7 z 13 z 10 z 5 6 z 14 z 25 z 2 z z > ---- - ----- - ----- - 4 a z - 10 z + -- - ----- - ----- + ---- - -- + 5 3 a 6 4 2 5 a a a a a a a 8 8 9 9 7 8 3 z 5 z z z > a z + 2 z + ---- + ---- + -- + -- 4 2 3 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 1 1 3 1 3 3 2 - + 5 q + ----- + ----- + ----- + ---- + --- + 4 q t + q t + 3 q t + q 7 3 5 2 3 2 3 q t q t q t q t q t 5 2 5 3 7 3 7 4 9 4 11 5 11 6 13 6 > 5 q t + 2 q t + 2 q t + 2 q t + 2 q t + 2 q t + q t + q t

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