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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X15,20,16,21 X14,8,15,7 X21,10,22,5 X18,11,19,12 X9,17,10,16 X22,17,11,18 X8,19,9,20 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-7 + 2q-6 - 4q-5 + 5q-4 - 5q-3 + 7q-2 - 4q-1 + 5 - 2q + q2

A2 (sl(3)) Invariant:

- q-22 - q-20 - q-18 - 3q-16 - q-14 - q-12 + 2q-10 + 6q-8 + 6q-6 + 8q-4 + 5q-2 + 4 + 3q2 + q6

HOMFLY-PT Polynomial:

2z-2 + 4 + 3z2 + z4 - 5a2z-2 - 10a2 - 10a2z2 - 5a2z4 - a2z6 + 4a4z-2 + 8a4 + 7a4z2 + 2a4z4 - a6z-2 - 2a6 - a6z2

Kauffman Polynomial:

a-2z2 + 2a-1z3 - 2z-2 + 7 - 9z2 + 5z4 + 5az-1 - 11az + 10az3 - 6az5 + 2az7 - 5a2z-2 + 15a2 - 27a2z2 + 24a2z4 - 13a2z6 + 3a2z8 + 9a3z-1 - 24a3z + 29a3z3 - 17a3z5 + a3z7 + a3z9 - 4a4z-2 + 12a4 - 22a4z2 + 30a4z4 - 22a4z6 + 5a4z8 + 5a5z-1 - 18a5z + 29a5z3 - 16a5z5 + a5z9 - a6z-2 + 3a6 - 5a6z2 + 11a6z4 - 9a6z6 + 2a6z8 + a7z-1 - 5a7z + 8a7z3 - 5a7z5 + a7z7

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 5                   1 j = 3                 2 1 j = 1               3     j = -1             3 4     j = -3           4 1       j = -5         2 4         j = -7       3 3           j = -9     1 2             j = -11   1 3               j = -13   1                 j = -15 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, 297]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 297]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[15, 20, 16, 21], X[14, 8, 15, 7], > X[21, 10, 22, 5], X[18, 11, 19, 12], X[9, 17, 10, 16], X[22, 17, 11, 18], > X[8, 19, 9, 20], 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]=	-7 2 4 5 5 7 4 2 5 - q + -- - -- + -- - -- + -- - - - 2 q + q 6 5 4 3 2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -20 -18 3 -14 -12 2 6 6 8 5 2 6 4 - q - q - q - --- - q - q + --- + -- + -- + -- + -- + 3 q + q 16 10 8 6 4 2 q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 297]][a, z]
Out[8]=	2 4 6 2 4 6 2 5 a 4 a a 2 2 2 4 2 4 - 10 a + 8 a - 2 a + -- - ---- + ---- - -- + 3 z - 10 a z + 7 a z - 2 2 2 2 z z z z 6 2 4 2 4 4 4 2 6 > a z + z - 5 a z + 2 a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 297]][a, z]
Out[9]=	2 4 6 3 5 7 2 4 6 2 5 a 4 a a 5 a 9 a 5 a a 7 + 15 a + 12 a + 3 a - -- - ---- - ---- - -- + --- + ---- + ---- + -- - 2 2 2 2 z z z z z z z z 2 3 5 7 2 z 2 2 4 2 > 11 a z - 24 a z - 18 a z - 5 a z - 9 z + -- - 27 a z - 22 a z - 2 a 3 6 2 2 z 3 3 3 5 3 7 3 4 > 5 a z + ---- + 10 a z + 29 a z + 29 a z + 8 a z + 5 z + a 2 4 4 4 6 4 5 3 5 5 5 7 5 > 24 a z + 30 a z + 11 a z - 6 a z - 17 a z - 16 a z - 5 a z - 2 6 4 6 6 6 7 3 7 7 7 2 8 > 13 a z - 22 a z - 9 a z + 2 a z + a z + a z + 3 a z + 4 8 6 8 3 9 5 9 > 5 a z + 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 1 1 1 3 1 2 3 3 - + 3 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + q 15 7 13 6 11 6 11 5 9 5 9 4 7 4 7 3 q t q t q t q t q t q t q t q t 2 4 4 1 3 3 3 2 5 2 > ----- + ----- + ----- + ---- + --- + 2 q t + q t + q t 5 3 5 2 3 2 3 q t q t q t q t q t

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