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DrawMorseLink

PD Presentation:

X10,1,11,2 X18,11,19,12 X6,9,7,10 X20,7,21,8 X8,19,1,20 X22,15,9,16 X12,4,13,3 X16,6,17,5 X4,14,5,13 X14,21,15,22 X2,18,3,17

Gauss Code:

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

Jones Polynomial:

q-15/2 - 4q-13/2 + 10q-11/2 - 16q-9/2 + 21q-7/2 - 25q-5/2 + 24q-3/2 - 22q-1/2 + 15q1/2 - 9q3/2 + 4q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-22 + 2q-20 - 3q-18 - q-16 + 2q-14 - 5q-12 + 5q-10 + 3q-6 + 5q-4 - 2q-2 + 6 - 3q2 + 2q6 - 2q8 + q10

HOMFLY-PT Polynomial:

- a-1z - 2a-1z3 - a-1z5 - 2az-1 - az + 3az3 + 3az5 + az7 + 3a3z-1 + 4a3z + 4a3z3 + 3a3z5 + a3z7 - a5z-1 - 2a5z - 2a5z3 - a5z5

Kauffman Polynomial:

a-3z3 - a-3z5 - a-2z2 + 5a-2z4 - 4a-2z6 + 2a-1z - 7a-1z3 + 12a-1z5 - 8a-1z7 + z2 - 7z4 + 14z6 - 10z8 + 2az-1 - az - 9az3 + 8az5 + 5az7 - 8az9 - 3a2 + 8a2z2 - 28a2z4 + 36a2z6 - 13a2z8 - 3a2z10 + 3a3z-1 - 7a3z + 10a3z3 - 20a3z5 + 31a3z7 - 16a3z9 - 3a4 + 13a4z2 - 31a4z4 + 37a4z6 - 11a4z8 - 3a4z10 + a5z-1 - 4a5z + 7a5z3 - 7a5z5 + 14a5z7 - 8a5z9 - a6 + 6a6z2 - 13a6z4 + 18a6z6 - 8a6z8 - 4a7z3 + 8a7z5 - 4a7z7 - a8z2 + 2a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                       1 j = 6                     3   j = 4                   6 1   j = 2                 9 3     j = 0               13 6       j = -2             13 11         j = -4           12 11           j = -6         9 13             j = -8       7 12               j = -10     3 9                 j = -12   1 7                   j = -14   3                     j = -16 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]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 309]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 309]]
Out[4]=	PD[X[10, 1, 11, 2], X[18, 11, 19, 12], X[6, 9, 7, 10], X[20, 7, 21, 8], > X[8, 19, 1, 20], X[22, 15, 9, 16], X[12, 4, 13, 3], X[16, 6, 17, 5], > X[4, 14, 5, 13], X[14, 21, 15, 22], X[2, 18, 3, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 7, -9, 8, -3, 4, -5}, > {3, -1, 2, -7, 9, -10, 6, -8, 11, -2, 5, -4, 10, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 4 10 16 21 25 24 22 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 15 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 7/2 > 9 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 2 3 -16 2 5 5 3 5 2 2 6 6 - q + --- - --- - q + --- - --- + --- + -- + -- - -- - 3 q + 2 q - 20 18 14 12 10 6 4 2 q q q q q q q q 8 10 > 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 309]][a, z]
Out[8]=	3 5 3 -2 a 3 a a z 3 5 2 z 3 3 3 ---- + ---- - -- - - - a z + 4 a z - 2 a z - ---- + 3 a z + 4 a z - z z z a a 5 5 3 z 5 3 5 5 5 7 3 7 > 2 a z - -- + 3 a z + 3 a z - a z + a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 309]][a, z]
Out[9]=	3 5 2 2 4 6 2 a 3 a a 2 z 3 5 2 z -3 a - 3 a - a + --- + ---- + -- + --- - a z - 7 a z - 4 a z + z - -- + z z z a 2 a 3 3 2 2 4 2 6 2 8 2 z 7 z 3 3 3 > 8 a z + 13 a z + 6 a z - a z + -- - ---- - 9 a z + 10 a z + 3 a a 4 5 3 7 3 4 5 z 2 4 4 4 6 4 > 7 a z - 4 a z - 7 z + ---- - 28 a z - 31 a z - 13 a z + 2 a 5 5 8 4 z 12 z 5 3 5 5 5 7 5 6 > 2 a z - -- + ----- + 8 a z - 20 a z - 7 a z + 8 a z + 14 z - 3 a a 6 7 4 z 2 6 4 6 6 6 8 6 8 z 7 3 7 > ---- + 36 a z + 37 a z + 18 a z - a z - ---- + 5 a z + 31 a z + 2 a a 5 7 7 7 8 2 8 4 8 6 8 9 > 14 a z - 4 a z - 10 z - 13 a z - 11 a z - 8 a z - 8 a z - 3 9 5 9 2 10 4 10 > 16 a z - 8 a z - 3 a z - 3 a z
In[10]:=	Kh[L][q, t]
Out[10]=	11 1 3 1 7 3 9 7 12 13 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 9 13 12 11 13 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 6 t + 9 q t + 3 q t + 6 q t + 6 3 6 2 4 2 4 2 q t q t q t q t q t 4 3 6 3 8 4 > q t + 3 q t + q t

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