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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-29/2 - 3q-27/2 + 5q-25/2 - 8q-23/2 + 10q-21/2 - 11q-19/2 + 11q-17/2 - 9q-15/2 + 6q-13/2 - 5q-11/2 + 2q-9/2 - q-7/2

A2 (sl(3)) Invariant:

- q-44 - q-42 + q-40 + 2q-36 + q-30 - 2q-28 + 3q-26 + 2q-22 + 2q-20 + 2q-16 - q-14 + q-12

HOMFLY-PT Polynomial:

- 3a7z - 7a7z3 - 5a7z5 - a7z7 - 2a9z-1 - 12a9z - 19a9z3 - 11a9z5 - 2a9z7 + 3a11z-1 + 14a11z + 13a11z3 + 3a11z5 - a13z-1 - 3a13z - a13z3

Kauffman Polynomial:

3a7z - 7a7z3 + 5a7z5 - a7z7 + a8z2 - 7a8z4 + 8a8z6 - 2a8z8 + 2a9z-1 - 13a9z + 23a9z3 - 23a9z5 + 14a9z7 - 3a9z9 - 3a10 + 20a10z2 - 35a10z4 + 22a10z6 - 2a10z8 - a10z10 + 3a11z-1 - 18a11z + 35a11z3 - 39a11z5 + 25a11z7 - 6a11z9 - 3a12 + 24a12z2 - 40a12z4 + 25a12z6 - 4a12z8 - a12z10 + a13z-1 - 3a13z + 3a13z3 - 5a13z5 + 6a13z7 - 3a13z9 - a14 + 4a14z2 - 7a14z4 + 7a14z6 - 4a14z8 - 2a15z + 2a15z3 + 3a15z5 - 4a15z7 + 4a16z4 - 4a16z6 - a17z + 4a17z3 - 3a17z5 + a18z2 - a18z4

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 = -6                       1 j = -8                     2 1 j = -10                   3     j = -12                 3 2     j = -14               6 3       j = -16             5 3         j = -18           6 6           j = -20         4 5             j = -22       4 6               j = -24     2 5                 j = -26   1 3                   j = -28   2                     j = -30 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, 273]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 273]]
Out[4]=	PD[X[10, 1, 11, 2], X[2, 11, 3, 12], X[12, 3, 13, 4], X[14, 5, 15, 6], > X[6, 9, 7, 10], X[18, 7, 19, 8], X[22, 19, 9, 20], X[20, 15, 21, 16], > X[16, 21, 17, 22], X[8, 17, 1, 18], X[4, 13, 5, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -2, 3, -11, 4, -5, 6, -10}, > {5, -1, 2, -3, 11, -4, 8, -9, 10, -6, 7, -8, 9, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(29/2) 3 5 8 10 11 11 9 6 q - ----- + ----- - ----- + ----- - ----- + ----- - ----- + ----- - 27/2 25/2 23/2 21/2 19/2 17/2 15/2 13/2 q q q q q q q q 5 2 -(7/2) > ----- + ---- - q 11/2 9/2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-44 -42 -40 2 -30 2 3 2 2 2 -14 -12 -q - q + q + --- + q - --- + --- + --- + --- + --- - q + q 36 28 26 22 20 16 q q q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 273]][a, z]
Out[8]=	9 11 13 -2 a 3 a a 7 9 11 13 7 3 ----- + ----- - --- - 3 a z - 12 a z + 14 a z - 3 a z - 7 a z - z z z 9 3 11 3 13 3 7 5 9 5 11 5 7 7 > 19 a z + 13 a z - a z - 5 a z - 11 a z + 3 a z - a z - 9 7 > 2 a z
In[9]:=	Kauffman[Link[11, Alternating, 273]][a, z]
Out[9]=	9 11 13 10 12 14 2 a 3 a a 7 9 11 -3 a - 3 a - a + ---- + ----- + --- + 3 a z - 13 a z - 18 a z - z z z 13 15 17 8 2 10 2 12 2 14 2 > 3 a z - 2 a z - a z + a z + 20 a z + 24 a z + 4 a z + 18 2 7 3 9 3 11 3 13 3 15 3 17 3 > a z - 7 a z + 23 a z + 35 a z + 3 a z + 2 a z + 4 a z - 8 4 10 4 12 4 14 4 16 4 18 4 7 5 > 7 a z - 35 a z - 40 a z - 7 a z + 4 a z - a z + 5 a z - 9 5 11 5 13 5 15 5 17 5 8 6 > 23 a z - 39 a z - 5 a z + 3 a z - 3 a z + 8 a z + 10 6 12 6 14 6 16 6 7 7 9 7 > 22 a z + 25 a z + 7 a z - 4 a z - a z + 14 a z + 11 7 13 7 15 7 8 8 10 8 12 8 > 25 a z + 6 a z - 4 a z - 2 a z - 2 a z - 4 a z - 14 8 9 9 11 9 13 9 10 10 12 10 > 4 a z - 3 a z - 6 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-8 -6 1 2 1 3 2 5 4 q + q + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 30 11 28 10 26 10 26 9 24 9 24 8 22 8 q t q t q t q t q t q t q t 6 4 5 6 6 5 3 6 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 22 7 20 7 20 6 18 6 18 5 16 5 16 4 14 4 q t q t q t q t q t q t q t q t 3 3 2 3 2 > ------ + ------ + ------ + ------ + ---- 14 3 12 3 12 2 10 2 8 q t q t q t q t q t

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