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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-4 + 3q-3 - 6q-2 + 11q-1 - 14 + 18q - 16q2 + 16q3 - 12q4 + 7q5 - 3q6 + q7

A2 (sl(3)) Invariant:

- q-12 - q-6 + 4q-4 + 4 + 5q2 + 2q4 + 8q6 + 4q10 - q14 + 3q16 - q18 + q20

HOMFLY-PT Polynomial:

a-4z-2 + 3a-4 + 6a-4z2 + 4a-4z4 + a-4z6 - 2a-2z-2 - 9a-2 - 18a-2z2 - 15a-2z4 - 6a-2z6 - a-2z8 + z-2 + 8 + 14z2 + 9z4 + 2z6 - 2a2 - 3a2z2 - a2z4

Kauffman Polynomial:

- a-8z2 + a-8z4 - 2a-7z3 + 3a-7z5 - a-6 + 4a-6z2 - 6a-6z4 + 6a-6z6 - a-5z + 11a-5z3 - 14a-5z5 + 9a-5z7 - a-4z-2 + 3a-4 - 6a-4z2 + 11a-4z4 - 15a-4z6 + 9a-4z8 + 2a-3z-1 - 8a-3z + 22a-3z3 - 22a-3z5 + a-3z7 + 5a-3z9 - 2a-2z-2 + 11a-2 - 37a-2z2 + 58a-2z4 - 51a-2z6 + 15a-2z8 + a-2z10 + 2a-1z-1 - 12a-1z + 20a-1z3 - 7a-1z5 - 15a-1z7 + 8a-1z9 - z-2 + 11 - 35z2 + 56z4 - 42z6 + 9z8 + z10 - 7az + 16az3 - 6az5 - 6az7 + 3az9 + 3a2 - 9a2z2 + 16a2z4 - 12a2z6 + 3a2z8 - 2a3z + 5a3z3 - 4a3z5 + a3z7

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 15                       1 j = 13                     2   j = 11                   5 1   j = 9                 8 3     j = 7               8 4       j = 5             8 8         j = 3           10 8           j = 1         7 11             j = -1       4 7               j = -3     2 7                 j = -5   1 4                   j = -7   2                     j = -9 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, 453]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 453]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 4, 15, 3], X[18, 11, 19, 12], X[16, 8, 17, 7], > X[20, 17, 21, 18], X[12, 19, 5, 20], X[8, 22, 9, 21], X[22, 10, 13, 9], > X[10, 14, 11, 13], X[2, 5, 3, 6], X[4, 16, 1, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, -7, 8, -9, 3, -6}, > {9, -2, 11, -4, 5, -3, 6, -5, 7, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-4 3 6 11 2 3 4 5 6 7 -14 - q + -- - -- + -- + 18 q - 16 q + 16 q - 12 q + 7 q - 3 q + q 3 2 q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-12 -6 4 2 4 6 10 14 16 18 20 4 - q - q + -- + 5 q + 2 q + 8 q + 4 q - q + 3 q - q + q 4 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 453]][a, z]
Out[8]=	2 2 3 9 2 -2 1 2 2 6 z 18 z 2 2 8 + -- - -- - 2 a + z + ----- - ----- + 14 z + ---- - ----- - 3 a z + 4 2 4 2 2 2 4 2 a a a z a z a a 4 4 6 6 8 4 4 z 15 z 2 4 6 z 6 z z > 9 z + ---- - ----- - a z + 2 z + -- - ---- - -- 4 2 4 2 2 a a a a a
In[9]:=	Kauffman[Link[11, Alternating, 453]][a, z]
Out[9]=	-6 3 11 2 -2 1 2 2 2 z 8 z 11 - a + -- + -- + 3 a - z - ----- - ----- + ---- + --- - -- - --- - 4 2 4 2 2 2 3 a z 5 3 a a a z a z a z a a 2 2 2 2 3 12 z 3 2 z 4 z 6 z 37 z 2 2 2 z > ---- - 7 a z - 2 a z - 35 z - -- + ---- - ---- - ----- - 9 a z - ---- + a 8 6 4 2 7 a a a a a 3 3 3 4 4 4 11 z 22 z 20 z 3 3 3 4 z 6 z 11 z > ----- + ----- + ----- + 16 a z + 5 a z + 56 z + -- - ---- + ----- + 5 3 a 8 6 4 a a a a a 4 5 5 5 5 58 z 2 4 3 z 14 z 22 z 7 z 5 3 5 6 > ----- + 16 a z + ---- - ----- - ----- - ---- - 6 a z - 4 a z - 42 z + 2 7 5 3 a a a a a 6 6 6 7 7 7 6 z 15 z 51 z 2 6 9 z z 15 z 7 3 7 > ---- - ----- - ----- - 12 a z + ---- + -- - ----- - 6 a z + a z + 6 4 2 5 3 a a a a a a 8 8 9 9 10 8 9 z 15 z 2 8 5 z 8 z 9 10 z > 9 z + ---- + ----- + 3 a z + ---- + ---- + 3 a z + z + --- 4 2 3 a 2 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	3 1 2 1 4 2 7 4 7 11 q + 10 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t q t q t q t q t q t q t q t 7 q 3 5 5 2 7 2 7 3 9 3 9 4 > --- + 8 q t + 8 q t + 8 q t + 8 q t + 4 q t + 8 q t + 3 q t + t 11 4 11 5 13 5 15 6 > 5 q t + q t + 2 q t + q t

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