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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X20,17,21,18 X18,13,19,14 X14,19,15,20 X4,21,1,22 X10,5,11,6 X12,3,13,4 X22,11,5,12 X2,9,3,10 X8,15,9,16

Gauss Code:

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

Jones Polynomial:

q-27/2 - 4q-25/2 + 9q-23/2 - 15q-21/2 + 22q-19/2 - 25q-17/2 + 25q-15/2 - 23q-13/2 + 16q-11/2 - 11q-9/2 + 4q-7/2 - q-5/2

A2 (sl(3)) Invariant:

- q-42 + 3q-38 - 3q-36 - q-34 + q-32 - 7q-30 + 2q-28 - q-26 + 2q-24 + 6q-22 - q-20 + 8q-18 - q-16 + 4q-12 - 3q-10 + q-8

HOMFLY-PT Polynomial:

- a5z3 - a5z5 - 3a7z-1 - 10a7z - 11a7z3 - 4a7z5 + 5a9z-1 + 7a9z - 2a9z5 - 2a11z-1 + 2a11z + 3a11z3 - a13z

Kauffman Polynomial:

a5z3 - a5z5 + 3a6z4 - 4a6z6 - 3a7z-1 + 10a7z - 16a7z3 + 17a7z5 - 10a7z7 + 5a8 - 7a8z2 - 3a8z4 + 16a8z6 - 12a8z8 - 5a9z-1 + 13a9z - 24a9z3 + 25a9z5 - 2a9z7 - 8a9z9 + 5a10 + 2a10z2 - 34a10z4 + 52a10z6 - 22a10z8 - 2a10z10 - 2a11z-1 + 4a11z - 14a11z3 + 12a11z5 + 15a11z7 - 14a11z9 + 14a12z2 - 41a12z4 + 48a12z6 - 17a12z8 - 2a12z10 + 2a13z - 13a13z3 + 14a13z5 + 3a13z7 - 6a13z9 - a14 + 4a14z2 - 11a14z4 + 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                     4 1 j = -8                   7     j = -10                 9 4     j = -12               14 7       j = -14             12 10         j = -16           13 13           j = -18         9 12             j = -20       6 13               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, 97]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 97]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[20, 17, 21, 18], X[18, 13, 19, 14], > X[14, 19, 15, 20], X[4, 21, 1, 22], X[10, 5, 11, 6], X[12, 3, 13, 4], > X[22, 11, 5, 12], X[2, 9, 3, 10], X[8, 15, 9, 16]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 8, -6}, {7, -1, 2, -11, 10, -7, 9, -8, 4, -5, 11, -2, 3, -4, > 5, -3, 6, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-(27/2) 4 9 15 22 25 25 23 16 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 11 4 -(5/2) > ---- + ---- - q 9/2 7/2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 3 3 -34 -32 7 2 -26 2 6 -20 8 -q + --- - --- - q + q - --- + --- - q + --- + --- - q + --- - 38 36 30 28 24 22 18 q q q q q q q -16 4 3 -8 > q + --- - --- + q 12 10 q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 97]][a, z]
Out[8]=	7 9 11 -3 a 5 a 2 a 7 9 11 13 5 3 7 3 ----- + ---- - ----- - 10 a z + 7 a z + 2 a z - a z - a z - 11 a z + z z z 11 3 5 5 7 5 9 5 > 3 a z - a z - 4 a z - 2 a z
In[9]:=	Kauffman[Link[11, Alternating, 97]][a, z]
Out[9]=	7 9 11 8 10 14 3 a 5 a 2 a 7 9 11 5 a + 5 a - a - ---- - ---- - ----- + 10 a z + 13 a z + 4 a z + z z z 13 15 8 2 10 2 12 2 14 2 16 2 > 2 a z + a z - 7 a z + 2 a z + 14 a z + 4 a z - a z + 5 3 7 3 9 3 11 3 13 3 15 3 6 4 > a z - 16 a z - 24 a z - 14 a z - 13 a z - 6 a z + 3 a z - 8 4 10 4 12 4 14 4 16 4 5 5 7 5 > 3 a z - 34 a z - 41 a z - 11 a z + 2 a z - a z + 17 a z + 9 5 11 5 13 5 15 5 6 6 8 6 > 25 a z + 12 a z + 14 a z + 9 a z - 4 a z + 16 a z + 10 6 12 6 14 6 16 6 7 7 9 7 > 52 a z + 48 a z + 15 a z - a z - 10 a z - 2 a z + 11 7 13 7 15 7 8 8 10 8 12 8 > 15 a z + 3 a z - 4 a z - 12 a z - 22 a z - 17 a z - 14 8 9 9 11 9 13 9 10 10 12 10 > 7 a z - 8 a z - 14 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 6 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 13 9 12 13 13 12 10 14 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 7 9 4 7 4 > ------ + ------ + ------ + ----- + ---- 12 3 10 3 10 2 8 2 6 q t q t q t q t q t

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