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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-10 + 3q-9 - 6q-8 + 11q-7 - 13q-6 + 17q-5 - 16q-4 + 15q-3 - 11q-2 + 7q-1 - 3 + q

A2 (sl(3)) Invariant:

- q-32 - q-30 + 2q-28 - q-26 + q-24 + 7q-22 + 2q-20 + 6q-18 + 5q-16 + q-14 + 3q-12 - 2q-10 + 3q-8 + q-6 - 2q-4 + 4q-2 - 1 - q2 + q4

HOMFLY-PT Polynomial:

z2 + 2a2 + 2a2z2 - a2z4 + a4z-2 - a4 - 3a4z2 - 3a4z4 - 2a6z-2 - 2a6 - a6z2 - 2a6z4 + a8z-2 + 2a8 + 3a8z2 - a10

Kauffman Polynomial:

- z2 + z4 - 2az3 + 3az5 - 2a2 + 6a2z2 - 7a2z4 + 6a2z6 - a3z + 8a3z3 - 11a3z5 + 8a3z7 - a4z-2 + 2a4 + 4a4z2 - 4a4z4 - 6a4z6 + 7a4z8 + 2a5z-1 - 10a5z + 22a5z3 - 24a5z5 + 4a5z7 + 4a5z9 - 2a6z-2 + 12a6 - 28a6z2 + 39a6z4 - 38a6z6 + 12a6z8 + a6z10 + 2a7z-1 - 14a7z + 25a7z3 - 13a7z5 - 11a7z7 + 7a7z9 - a8z-2 + 12a8 - 34a8z2 + 51a8z4 - 38a8z6 + 8a8z8 + a8z10 - 7a9z + 18a9z3 - 7a9z5 - 6a9z7 + 3a9z9 + 3a10 - 9a10z2 + 16a10z4 - 12a10z6 + 3a10z8 - 2a11z + 5a11z3 - 4a11z5 + a11z7

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 3                       1 j = 1                     2   j = -1                   5 1   j = -3                 7 3     j = -5               8 4       j = -7             9 8         j = -9           8 7           j = -11         6 10             j = -13       5 7               j = -15     2 7                 j = -17   1 4                   j = -19   2                     j = -21 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, 469]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 469]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[18, 11, 19, 12], X[16, 8, 17, 7], > X[8, 16, 9, 15], X[22, 17, 15, 18], X[12, 21, 13, 22], X[20, 13, 21, 14], > X[14, 19, 5, 20], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {5, -4, 6, -3, 9, -8, 7, -6}, > {10, -1, 4, -5, 11, -2, 3, -7, 8, -9}]
In[6]:=	Jones[L][q]
Out[6]=	-10 3 6 11 13 17 16 15 11 7 -3 - q + -- - -- + -- - -- + -- - -- + -- - -- + - + q 9 8 7 6 5 4 3 2 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 -30 2 -26 -24 7 2 6 5 -14 3 -1 - q - q + --- - q + q + --- + --- + --- + --- + q + --- - 28 22 20 18 16 12 q q q q q q 2 3 -6 2 4 2 4 > --- + -- + q - -- + -- - q + q 10 8 4 2 q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 469]][a, z]
Out[8]=	4 6 8 2 4 6 8 10 a 2 a a 2 2 2 4 2 2 a - a - 2 a + 2 a - a + -- - ---- + -- + z + 2 a z - 3 a z - 2 2 2 z z z 6 2 8 2 2 4 4 4 6 4 > a z + 3 a z - a z - 3 a z - 2 a z
In[9]:=	Kauffman[Link[11, Alternating, 469]][a, z]
Out[9]=	4 6 8 5 7 2 4 6 8 10 a 2 a a 2 a 2 a 3 -2 a + 2 a + 12 a + 12 a + 3 a - -- - ---- - -- + ---- + ---- - a z - 2 2 2 z z z z z 5 7 9 11 2 2 2 4 2 6 2 > 10 a z - 14 a z - 7 a z - 2 a z - z + 6 a z + 4 a z - 28 a z - 8 2 10 2 3 3 3 5 3 7 3 9 3 > 34 a z - 9 a z - 2 a z + 8 a z + 22 a z + 25 a z + 18 a z + 11 3 4 2 4 4 4 6 4 8 4 10 4 > 5 a z + z - 7 a z - 4 a z + 39 a z + 51 a z + 16 a z + 5 3 5 5 5 7 5 9 5 11 5 2 6 > 3 a z - 11 a z - 24 a z - 13 a z - 7 a z - 4 a z + 6 a z - 4 6 6 6 8 6 10 6 3 7 5 7 7 7 > 6 a z - 38 a z - 38 a z - 12 a z + 8 a z + 4 a z - 11 a z - 9 7 11 7 4 8 6 8 8 8 10 8 5 9 > 6 a z + a z + 7 a z + 12 a z + 8 a z + 3 a z + 4 a z + 7 9 9 9 6 10 8 10 > 7 a z + 3 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 5 1 2 1 4 2 7 5 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 21 9 19 8 17 8 17 7 15 7 15 6 13 6 q q t q t q t q t q t q t q t 7 6 10 8 7 9 8 8 4 > ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 13 5 11 5 11 4 9 4 9 3 7 3 7 2 5 2 5 q t q t q t q t q t q t q t q t q t 7 t 3 2 > ---- + - + 2 q t + q t 3 q q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a469

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