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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X20,14,21,13 X14,7,15,8 X8,15,9,16 X18,11,5,12 X12,20,13,19 X16,22,17,21 X22,18,19,17 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

- q-7 + 3q-6 - 7q-5 + 12q-4 - 15q-3 + 18q-2 - 17q-1 + 17 - 11q + 7q2 - 3q3 + q4

A2 (sl(3)) Invariant:

- q-22 - q-20 - 4q-16 + q-14 + q-12 + 7q-8 + 2q-6 + 7q-4 + 5q-2 + 3 + 6q2 - 2q4 + 2q6 + q8 - q10 + q12

HOMFLY-PT Polynomial:

a-2 + 2a-2z2 + a-2z4 + 2z-2 + 4 - 2z4 - z6 - 5a2z-2 - 13a2 - 14a2z2 - 8a2z4 - 2a2z6 + 4a4z-2 + 10a4 + 9a4z2 + 3a4z4 - a6z-2 - 2a6 - a6z2

Kauffman Polynomial:

- a-4z2 + a-4z4 - 2a-3z3 + 3a-3z5 - 2a-2 + 6a-2z2 - 7a-2z4 + 6a-2z6 + 4a-1z3 - 9a-1z5 + 8a-1z7 - 2z-2 + 7 - 8z2 + 12z4 - 15z6 + 9z8 + 5az-1 - 22az + 42az3 - 35az5 + 4az7 + 5az9 - 5a2z-2 + 21a2 - 43a2z2 + 63a2z4 - 57a2z6 + 17a2z8 + a2z10 + 9a3z-1 - 39a3z + 64a3z3 - 37a3z5 - 8a3z7 + 8a3z9 - 4a4z-2 + 16a4 - 36a4z2 + 57a4z4 - 47a4z6 + 11a4z8 + a4z10 + 5a5z-1 - 21a5z + 34a5z3 - 18a5z5 - 3a5z7 + 3a5z9 - a6z-2 + 3a6 - 8a6z2 + 14a6z4 - 11a6z6 + 3a6z8 + a7z-1 - 4a7z + 6a7z3 - 4a7z5 + a7z7

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 9                       1 j = 7                     3 1 j = 5                   4     j = 3                 7 3     j = 1               10 4       j = -1             9 9         j = -3           9 8           j = -5         6 9             j = -7       6 9               j = -9     2 7                 j = -11   1 5                   j = -13   2                     j = -15 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 489]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 489]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[20, 14, 21, 13], X[14, 7, 15, 8], > X[8, 15, 9, 16], X[18, 11, 5, 12], X[12, 20, 13, 19], X[16, 22, 17, 21], > X[22, 18, 19, 17], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {7, -3, 8, -9}, > {10, -1, 4, -5, 11, -2, 6, -7, 3, -4, 5, -8, 9, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-7 3 7 12 15 18 17 2 3 4 17 - q + -- - -- + -- - -- + -- - -- - 11 q + 7 q - 3 q + q 6 5 4 3 2 q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-22 -20 4 -14 -12 7 2 7 5 2 4 6 3 - q - q - --- + q + q + -- + -- + -- + -- + 6 q - 2 q + 2 q + 16 8 6 4 2 q q q q q 8 10 12 > q - q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 489]][a, z]
Out[8]=	2 4 6 2 -2 2 4 6 2 5 a 4 a a 2 z 2 2 4 + a - 13 a + 10 a - 2 a + -- - ---- + ---- - -- + ---- - 14 a z + 2 2 2 2 2 z z z z a 4 4 2 6 2 4 z 2 4 4 4 6 2 6 > 9 a z - a z - 2 z + -- - 8 a z + 3 a z - z - 2 a z 2 a
In[9]:=	Kauffman[Link[11, Alternating, 489]][a, z]
Out[9]=	2 4 6 3 5 2 2 4 6 2 5 a 4 a a 5 a 9 a 5 a 7 - -- + 21 a + 16 a + 3 a - -- - ---- - ---- - -- + --- + ---- + ---- + 2 2 2 2 2 z z z a z z z z 7 2 2 a 3 5 7 2 z 6 z 2 2 > -- - 22 a z - 39 a z - 21 a z - 4 a z - 8 z - -- + ---- - 43 a z - z 4 2 a a 3 3 4 2 6 2 2 z 4 z 3 3 3 5 3 > 36 a z - 8 a z - ---- + ---- + 42 a z + 64 a z + 34 a z + 3 a a 4 4 5 7 3 4 z 7 z 2 4 4 4 6 4 3 z > 6 a z + 12 z + -- - ---- + 63 a z + 57 a z + 14 a z + ---- - 4 2 3 a a a 5 6 9 z 5 3 5 5 5 7 5 6 6 z 2 6 > ---- - 35 a z - 37 a z - 18 a z - 4 a z - 15 z + ---- - 57 a z - a 2 a 7 4 6 6 6 8 z 7 3 7 5 7 7 7 8 > 47 a z - 11 a z + ---- + 4 a z - 8 a z - 3 a z + a z + 9 z + a 2 8 4 8 6 8 9 3 9 5 9 2 10 4 10 > 17 a z + 11 a z + 3 a z + 5 a z + 8 a z + 3 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	9 1 2 1 5 2 7 6 9 - + 10 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + q 15 7 13 6 11 6 11 5 9 5 9 4 7 4 7 3 q t q t q t q t q t q t q t q t 6 9 9 8 9 3 3 2 5 2 > ----- + ----- + ----- + ---- + --- + 4 q t + 7 q t + 3 q t + 4 q t + 5 3 5 2 3 2 3 q t q t q t q t q t 7 3 7 4 9 4 > 3 q t + q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a489

L11a488 L11a490