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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X15,18,16,11 X20,9,21,10 X22,13,19,14 X14,21,15,22 X10,19,5,20 X17,8,18,9 X7,16,8,17 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

q-23/2 - 5q-21/2 + 6q-19/2 - 11q-17/2 + 10q-15/2 - 12q-13/2 + 8q-11/2 - 7q-9/2 + 3q-7/2 - q-5/2

A2 (sl(3)) Invariant:

q-38 + 2q-36 + 5q-34 + 11q-32 + 10q-30 + 13q-28 + 11q-26 + 7q-24 + 9q-22 + 3q-20 + 6q-18 + 2q-16 + 2q-12 - 2q-10 + q-8

HOMFLY-PT Polynomial:

- 2a5z3 - a5z5 - a7z-3 - 5a7z-1 - 12a7z - 11a7z3 - 3a7z5 + 3a9z-3 + 10a9z-1 + 14a9z + 5a9z3 - 3a11z-3 - 5a11z-1 - 2a11z + a13z-3

Kauffman Polynomial:

2a5z3 - a5z5 + 5a6z4 - 3a6z6 + a7z-3 - 5a7z-1 + 12a7z - 19a7z3 + 17a7z5 - 6a7z7 - 3a8z-2 + 10a8 - 14a8z2 + 5a8z4 + 5a8z6 - 4a8z8 + 3a9z-3 - 12a9z-1 + 23a9z - 29a9z3 + 22a9z5 - 7a9z7 - a9z9 - 6a10z-2 + 19a10 - 20a10z2 + 4a10z4 + 6a10z6 - 5a10z8 + 3a11z-3 - 12a11z-1 + 15a11z - 3a11z3 - a11z5 - a11z7 - a11z9 - 3a12z-2 + 10a12 - 6a12z2 + 3a12z4 - 2a12z6 - a12z8 + a13z-3 - 5a13z-1 + 4a13z + 5a13z3 - 5a13z5 - a14z4

Khovanov Homology:

trqj 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             4 3     j = -12           8 4       j = -14         4 6         j = -16       7 6           j = -18     3 8             j = -20   2 3               j = -22   4                 j = -24 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]=	4
In[3]:=	Show[DrawMorseLink[Link[11, NonAlternating, 457]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 457]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[15, 18, 16, 11], X[20, 9, 21, 10], > X[22, 13, 19, 14], X[14, 21, 15, 22], X[10, 19, 5, 20], X[17, 8, 18, 9], > X[7, 16, 8, 17], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {7, -4, 6, -5}, {10, -1, -9, 8, 4, -7}, > {11, -2, 5, -6, -3, 9, -8, 3}]
In[6]:=	Jones[L][q]
Out[6]=	-(23/2) 5 6 11 10 12 8 7 3 -(5/2) q - ----- + ----- - ----- + ----- - ----- + ----- - ---- + ---- - q 21/2 19/2 17/2 15/2 13/2 11/2 9/2 7/2 q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-38 2 5 11 10 13 11 7 9 3 6 2 2 q + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- - 36 34 32 30 28 26 24 22 20 18 16 12 q q q q q q q q q q q q 2 -8 > --- + q 10 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 457]][a, z]
Out[8]=	7 9 11 13 7 9 11 a 3 a 3 a a 5 a 10 a 5 a 7 9 -(--) + ---- - ----- + --- - ---- + ----- - ----- - 12 a z + 14 a z - 3 3 3 3 z z z z z z z 11 5 3 7 3 9 3 5 5 7 5 > 2 a z - 2 a z - 11 a z + 5 a z - a z - 3 a z
In[9]:=	Kauffman[Link[11, NonAlternating, 457]][a, z]
Out[9]=	7 9 11 13 8 10 12 8 10 12 a 3 a 3 a a 3 a 6 a 3 a 10 a + 19 a + 10 a + -- + ---- + ----- + --- - ---- - ----- - ----- - 3 3 3 3 2 2 2 z z z z z z z 7 9 11 13 5 a 12 a 12 a 5 a 7 9 11 13 > ---- - ----- - ------ - ----- + 12 a z + 23 a z + 15 a z + 4 a z - z z z z 8 2 10 2 12 2 5 3 7 3 9 3 > 14 a z - 20 a z - 6 a z + 2 a z - 19 a z - 29 a z - 11 3 13 3 6 4 8 4 10 4 12 4 14 4 > 3 a z + 5 a z + 5 a z + 5 a z + 4 a z + 3 a z - a z - 5 5 7 5 9 5 11 5 13 5 6 6 8 6 > a z + 17 a z + 22 a z - a z - 5 a z - 3 a z + 5 a z + 10 6 12 6 7 7 9 7 11 7 8 8 10 8 > 6 a z - 2 a z - 6 a z - 7 a z - a z - 4 a z - 5 a z - 12 8 9 9 11 9 > a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 4 2 3 3 8 7 q + q + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 24 9 22 8 20 8 20 7 18 7 18 6 16 6 q t q t q t q t q t q t q t 6 4 6 8 4 4 3 4 3 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ---- 16 5 14 5 14 4 12 4 12 3 10 3 10 2 8 2 6 q t q t q t q t q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n457

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