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DrawMorseLink

PD Presentation:

X8192 X12,3,13,4 X20,13,21,14 X16,9,17,10 X10,19,11,20 X22,15,7,16 X14,21,15,22 X18,5,19,6 X2738 X4,11,5,12 X6,17,1,18

Gauss Code:

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

Jones Polynomial:

q-27/2 - 3q-25/2 + 7q-23/2 - 11q-21/2 + 15q-19/2 - 18q-17/2 + 17q-15/2 - 16q-13/2 + 11q-11/2 - 7q-9/2 + 3q-7/2 - q-5/2

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

- 2a5z3 - a5z5 - 4a7z - 8a7z3 - 3a7z5 - 3a9z-1 - 8a9z - 9a9z3 - 3a9z5 + 5a11z-1 + 10a11z + 4a11z3 - 2a13z-1 - a13z

Kauffman Polynomial:

2a5z3 - a5z5 + 5a6z4 - 3a6z6 + 4a7z - 12a7z3 + 14a7z5 - 6a7z7 + 3a8z2 - 15a8z4 + 16a8z6 - 7a8z8 + 3a9z-1 - 11a9z + 18a9z3 - 23a9z5 + 15a9z7 - 6a9z9 - 5a10 + 21a10z2 - 36a10z4 + 23a10z6 - 5a10z8 - 2a10z10 + 5a11z-1 - 22a11z + 47a11z3 - 51a11z5 + 31a11z7 - 10a11z9 - 5a12 + 17a12z2 - 17a12z4 + 12a12z6 - 2a12z8 - 2a12z10 + 2a13z-1 - 7a13z + 10a13z3 - 5a13z5 + 7a13z7 - 4a13z9 - 4a14z2 + 2a14z4 + 7a14z6 - 4a14z8 - 5a15z3 + 8a15z5 - 3a15z7 + a16 - 3a16z2 + 3a16z4 - 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                   4     j = -10                 7 3     j = -12               9 4       j = -14             8 7         j = -16           10 9           j = -18         6 9             j = -20       5 9               j = -22     2 6                 j = -24   1 5                   j = -26   2                     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, 235]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 235]]
Out[4]=	PD[X[8, 1, 9, 2], X[12, 3, 13, 4], X[20, 13, 21, 14], X[16, 9, 17, 10], > X[10, 19, 11, 20], X[22, 15, 7, 16], X[14, 21, 15, 22], X[18, 5, 19, 6], > X[2, 7, 3, 8], X[4, 11, 5, 12], X[6, 17, 1, 18]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -9, 2, -10, 8, -11}, > {9, -1, 4, -5, 10, -2, 3, -7, 6, -4, 11, -8, 5, -3, 7, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(27/2) 3 7 11 15 18 17 16 11 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 7 3 -(5/2) > ---- + ---- - q 9/2 7/2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 2 3 3 -30 5 -26 2 3 2 4 -16 -q - --- - --- + --- - q + --- + q + --- + --- - --- + --- - q + 40 36 32 28 24 22 20 18 q q q q q q q q 2 2 -8 > --- - --- + q 12 10 q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 235]][a, z]
Out[8]=	9 11 13 -3 a 5 a 2 a 7 9 11 13 5 3 ----- + ----- - ----- - 4 a z - 8 a z + 10 a z - a z - 2 a z - z z z 7 3 9 3 11 3 5 5 7 5 9 5 > 8 a z - 9 a z + 4 a z - a z - 3 a z - 3 a z
In[9]:=	Kauffman[Link[11, Alternating, 235]][a, z]
Out[9]=	9 11 13 10 12 16 3 a 5 a 2 a 7 9 11 -5 a - 5 a + a + ---- + ----- + ----- + 4 a z - 11 a z - 22 a z - z z z 13 8 2 10 2 12 2 14 2 16 2 5 3 > 7 a z + 3 a z + 21 a z + 17 a z - 4 a z - 3 a z + 2 a z - 7 3 9 3 11 3 13 3 15 3 6 4 > 12 a z + 18 a z + 47 a z + 10 a z - 5 a z + 5 a z - 8 4 10 4 12 4 14 4 16 4 5 5 7 5 > 15 a z - 36 a z - 17 a z + 2 a z + 3 a z - a z + 14 a z - 9 5 11 5 13 5 15 5 6 6 8 6 > 23 a z - 51 a z - 5 a z + 8 a z - 3 a z + 16 a z + 10 6 12 6 14 6 16 6 7 7 9 7 > 23 a z + 12 a z + 7 a z - a z - 6 a z + 15 a z + 11 7 13 7 15 7 8 8 10 8 12 8 > 31 a z + 7 a z - 3 a z - 7 a z - 5 a z - 2 a z - 14 8 9 9 11 9 13 9 10 10 12 10 > 4 a z - 6 a z - 10 a z - 4 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 2 1 5 2 6 5 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 9 6 9 10 9 8 7 9 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 4 7 3 4 3 > ------ + ------ + ------ + ----- + ---- 12 3 10 3 10 2 8 2 6 q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a235

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