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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-25/2 - 2q-23/2 + 3q-21/2 - 4q-19/2 + 4q-17/2 - 5q-15/2 + 5q-13/2 - 4q-11/2 + 3q-9/2 - 3q-7/2 + q-5/2 - q-3/2

A2 (sl(3)) Invariant:

- q-36 - q-32 + q-26 + 2q-24 + q-20 - q-18 + q-16 + q-14 + 2q-12 + 2q-10 + q-8 + q-6

HOMFLY-PT Polynomial:

- 2a5z-1 - 13a5z - 16a5z3 - 7a5z5 - a5z7 + 3a7z-1 + 18a7z + 30a7z3 + 23a7z5 + 8a7z7 + a7z9 - a9z-1 - 7a9z - 11a9z3 - 6a9z5 - a9z7

Kauffman Polynomial:

2a5z-1 - 15a5z + 29a5z3 - 23a5z5 + 8a5z7 - a5z9 - 3a6 + 11a6z2 - 3a6z4 - 9a6z6 + 6a6z8 - a6z10 + 3a7z-1 - 22a7z + 56a7z3 - 61a7z5 + 27a7z7 - 4a7z9 - 3a8 + 23a8z2 - 34a8z4 + 12a8z6 + 2a8z8 - a8z10 + a9z-1 - 6a9z + 13a9z3 - 22a9z5 + 15a9z7 - 3a9z9 - a10 + 7a10z2 - 19a10z4 + 17a10z6 - 4a10z8 - a11z - 5a11z3 + 12a11z5 - 4a11z7 - 2a12z2 + 9a12z4 - 4a12z6 - 2a13z + 7a13z3 - 4a13z5 + 2a14z2 - 3a14z4 - 2a15z3 - a16z2

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = -2                       1 j = -4                         j = -6                   3 1   j = -8                 1 1     j = -10               3 2       j = -12             2 1         j = -14           3 3           j = -16         2 3             j = -18       2 2               j = -20     1 2                 j = -22   1 2                   j = -24   1                     j = -26 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, 277]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 277]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[14, 5, 15, 6], > X[8, 9, 1, 10], X[4, 13, 5, 14], X[20, 15, 21, 16], X[18, 8, 19, 7], > X[6, 20, 7, 19], X[22, 17, 9, 18], X[16, 21, 17, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -6, 4, -9, 8, -5}, > {5, -1, 2, -3, 6, -4, 7, -11, 10, -8, 9, -7, 11, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(25/2) 2 3 4 4 5 5 4 3 q - ----- + ----- - ----- + ----- - ----- + ----- - ----- + ---- - 23/2 21/2 19/2 17/2 15/2 13/2 11/2 9/2 q q q q q q q q 3 -(5/2) -(3/2) > ---- + q - q 7/2 q
In[7]:=	A2Invariant[L][q]
Out[7]=	-36 -32 -26 2 -20 -18 -16 -14 2 2 -8 -6 -q - q + q + --- + q - q + q + q + --- + --- + q + q 24 12 10 q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 277]][a, z]
Out[8]=	5 7 9 -2 a 3 a a 5 7 9 5 3 7 3 ----- + ---- - -- - 13 a z + 18 a z - 7 a z - 16 a z + 30 a z - z z z 9 3 5 5 7 5 9 5 5 7 7 7 9 7 7 9 > 11 a z - 7 a z + 23 a z - 6 a z - a z + 8 a z - a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 277]][a, z]
Out[9]=	5 7 9 6 8 10 2 a 3 a a 5 7 9 11 -3 a - 3 a - a + ---- + ---- + -- - 15 a z - 22 a z - 6 a z - a z - z z z 13 6 2 8 2 10 2 12 2 14 2 16 2 > 2 a z + 11 a z + 23 a z + 7 a z - 2 a z + 2 a z - a z + 5 3 7 3 9 3 11 3 13 3 15 3 6 4 > 29 a z + 56 a z + 13 a z - 5 a z + 7 a z - 2 a z - 3 a z - 8 4 10 4 12 4 14 4 5 5 7 5 > 34 a z - 19 a z + 9 a z - 3 a z - 23 a z - 61 a z - 9 5 11 5 13 5 6 6 8 6 10 6 > 22 a z + 12 a z - 4 a z - 9 a z + 12 a z + 17 a z - 12 6 5 7 7 7 9 7 11 7 6 8 8 8 > 4 a z + 8 a z + 27 a z + 15 a z - 4 a z + 6 a z + 2 a z - 10 8 5 9 7 9 9 9 6 10 8 10 > 4 a z - a z - 4 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-8 3 1 1 1 2 1 2 2 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 6 26 9 24 8 22 8 22 7 20 7 20 6 18 6 q q t q t q t q t q t q t q t 2 2 3 3 3 2 1 3 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 18 5 16 5 16 4 14 4 14 3 12 3 12 2 10 2 q t q t q t q t q t q t q t q t 2 2 1 t t > ----- + ---- + -- + -- 10 8 6 2 q t q t q q

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