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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-27/2 - 4q-25/2 + 9q-23/2 - 15q-21/2 + 19q-19/2 - 22q-17/2 + 22q-15/2 - 19q-13/2 + 13q-11/2 - 8q-9/2 + 3q-7/2 - q-5/2

A2 (sl(3)) Invariant:

- q-42 - q-40 + 2q-38 - 3q-36 + q-34 + 4q-32 - 2q-30 + 5q-28 - q-26 + q-24 + 2q-22 - 3q-20 + 5q-18 - 2q-16 + 3q-12 - 2q-10 + q-8

HOMFLY-PT Polynomial:

- a5z - 2a5z3 - a5z5 - 4a7z - 7a7z3 - 3a7z5 - 2a9z-1 - 5a9z - 7a9z3 - 3a9z5 + 3a11z-1 + 7a11z + 4a11z3 - a13z-1 - a13z

Kauffman Polynomial:

- a5z + 2a5z3 - a5z5 - a6z2 + 4a6z4 - 3a6z6 + 4a7z - 9a7z3 + 10a7z5 - 6a7z7 - a8z2 - 3a8z4 + 8a8z6 - 7a8z8 + 2a9z-1 - 8a9z + 8a9z3 - 4a9z5 + 5a9z7 - 6a9z9 - 3a10 + 12a10z2 - 28a10z4 + 31a10z6 - 12a10z8 - 2a10z10 + 3a11z-1 - 14a11z + 19a11z3 - 16a11z5 + 21a11z7 - 12a11z9 - 3a12 + 15a12z2 - 31a12z4 + 36a12z6 - 12a12z8 - 2a12z10 + a13z-1 - 6a13z3 + 8a13z5 + 6a13z7 - 6a13z9 - a14 + 2a14z2 - 8a14z4 + 15a14z6 - 7a14z8 + a15z - 6a15z3 + 9a15z5 - 4a15z7 - a16z2 + 2a16z4 - a16z6

Khovanov Homology:

trqj r = -11 r = -10 r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -4                       1 j = -6                     3 1 j = -8                   5     j = -10                 8 3     j = -12               11 5       j = -14             11 8         j = -16           11 11           j = -18         8 11             j = -20       7 11               j = -22     3 9                 j = -24   1 6                   j = -26   3                     j = -28 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, 337]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 337]]
Out[4]=	PD[X[10, 1, 11, 2], X[22, 5, 9, 6], X[14, 3, 15, 4], X[4, 15, 5, 16], > X[16, 7, 17, 8], X[20, 17, 21, 18], X[18, 11, 19, 12], X[12, 19, 13, 20], > X[2, 9, 3, 10], X[8, 13, 1, 14], X[6, 21, 7, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 3, -4, 2, -11, 5, -10}, > {9, -1, 7, -8, 10, -3, 4, -5, 6, -7, 8, -6, 11, -2}]
In[6]:=	Jones[L][q]
Out[6]=	-(27/2) 4 9 15 19 22 22 19 13 q - ----- + ----- - ----- + ----- - ----- + ----- - ----- + ----- - 25/2 23/2 21/2 19/2 17/2 15/2 13/2 11/2 q q q q q q q q 8 3 -(5/2) > ---- + ---- - q 9/2 7/2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 -40 2 3 -34 4 2 5 -26 -24 2 3 -q - q + --- - --- + q + --- - --- + --- - q + q + --- - --- + 38 36 32 30 28 22 20 q q q q q q q 5 2 3 2 -8 > --- - --- + --- - --- + q 18 16 12 10 q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 337]][a, z]
Out[8]=	9 11 13 -2 a 3 a a 5 7 9 11 13 5 3 ----- + ----- - --- - a z - 4 a z - 5 a z + 7 a z - a z - 2 a z - z z z 7 3 9 3 11 3 5 5 7 5 9 5 > 7 a z - 7 a z + 4 a z - a z - 3 a z - 3 a z
In[9]:=	Kauffman[Link[11, Alternating, 337]][a, z]
Out[9]=	9 11 13 10 12 14 2 a 3 a a 5 7 9 11 -3 a - 3 a - a + ---- + ----- + --- - a z + 4 a z - 8 a z - 14 a z + z z z 15 6 2 8 2 10 2 12 2 14 2 16 2 > a z - a z - a z + 12 a z + 15 a z + 2 a z - a z + 5 3 7 3 9 3 11 3 13 3 15 3 6 4 > 2 a z - 9 a z + 8 a z + 19 a z - 6 a z - 6 a z + 4 a z - 8 4 10 4 12 4 14 4 16 4 5 5 7 5 > 3 a z - 28 a z - 31 a z - 8 a z + 2 a z - a z + 10 a z - 9 5 11 5 13 5 15 5 6 6 8 6 10 6 > 4 a z - 16 a z + 8 a z + 9 a z - 3 a z + 8 a z + 31 a z + 12 6 14 6 16 6 7 7 9 7 11 7 13 7 > 36 a z + 15 a z - a z - 6 a z + 5 a z + 21 a z + 6 a z - 15 7 8 8 10 8 12 8 14 8 9 9 > 4 a z - 7 a z - 12 a z - 12 a z - 7 a z - 6 a z - 11 9 13 9 10 10 12 10 > 12 a z - 6 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 3 1 6 3 9 7 q + q + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 28 11 26 10 24 10 24 9 22 9 22 8 20 8 q t q t q t q t q t q t q t 11 8 11 11 11 11 8 11 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 20 7 18 7 18 6 16 6 16 5 14 5 14 4 12 4 q t q t q t q t q t q t q t q t 5 8 3 5 3 > ------ + ------ + ------ + ----- + ---- 12 3 10 3 10 2 8 2 6 q t q t q t q t q t

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