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DrawMorseLink

PD Presentation:

X6172 X12,4,13,3 X14,5,15,6 X20,11,21,12 X22,17,11,18 X16,21,17,22 X10,13,5,14 X19,8,20,9 X7,18,8,19 X2,9,3,10 X4,16,1,15

Gauss Code:

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

Jones Polynomial:

- q-11 + 4q-10 - 8q-9 + 11q-8 - 14q-7 + 15q-6 - 12q-5 + 11q-4 - 5q-3 + 3q-2

A2 (sl(3)) Invariant:

- q-34 + q-30 - 4q-28 - 2q-24 - q-22 + 5q-20 + 2q-18 + 9q-16 + 4q-14 + 5q-12 + 6q-10 + 3q-6

HOMFLY-PT Polynomial:

2a4z-2 + 8a4 + 9a4z2 + 3a4z4 - 5a6z-2 - 15a6 - 17a6z2 - 9a6z4 - 2a6z6 + 4a8z-2 + 8a8 + 8a8z2 + 3a8z4 - a10z-2 - a10 - a10z2

Kauffman Polynomial:

- 2a4z-2 + 10a4 - 14a4z2 + 6a4z4 + 5a5z-1 - 16a5z + 13a5z3 - 6a5z5 + 3a5z7 - 5a6z-2 + 20a6 - 37a6z2 + 33a6z4 - 15a6z6 + 5a6z8 + 9a7z-1 - 27a7z + 33a7z3 - 20a7z5 + 5a7z7 + 2a7z9 - 4a8z-2 + 13a8 - 27a8z2 + 35a8z4 - 27a8z6 + 11a8z8 + 5a9z-1 - 13a9z + 27a9z3 - 28a9z5 + 9a9z7 + 2a9z9 - a10z-2 + 2a10 - 3a10z2 + 2a10z4 - 8a10z6 + 6a10z8 + a11z-1 - 2a11z + 6a11z3 - 13a11z5 + 7a11z7 + a12z2 - 6a12z4 + 4a12z6 - a13z3 + a13z5

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -3                   3 j = -5                 4 2 j = -7               7 1   j = -9             6 5     j = -11           9 6       j = -13         5 6         j = -15       6 9           j = -17     3 6             j = -19   1 5               j = -21   3                 j = -23 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, NonAlternating, 318]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 318]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 4, 13, 3], X[14, 5, 15, 6], X[20, 11, 21, 12], > X[22, 17, 11, 18], X[16, 21, 17, 22], X[10, 13, 5, 14], X[19, 8, 20, 9], > X[7, 18, 8, 19], X[2, 9, 3, 10], X[4, 16, 1, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {3, -1, -9, 8, 10, -7}, > {4, -2, 7, -3, 11, -6, 5, 9, -8, -4, 6, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-11 4 8 11 14 15 12 11 5 3 -q + --- - -- + -- - -- + -- - -- + -- - -- + -- 10 9 8 7 6 5 4 3 2 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 -30 4 2 -22 5 2 9 4 5 6 3 -q + q - --- - --- - q + --- + --- + --- + --- + --- + --- + -- 28 24 20 18 16 14 12 10 6 q q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 318]][a, z]
Out[8]=	4 6 8 10 4 6 8 10 2 a 5 a 4 a a 4 2 6 2 8 a - 15 a + 8 a - a + ---- - ---- + ---- - --- + 9 a z - 17 a z + 2 2 2 2 z z z z 8 2 10 2 4 4 6 4 8 4 6 6 > 8 a z - a z + 3 a z - 9 a z + 3 a z - 2 a z
In[9]:=	Kauffman[Link[11, NonAlternating, 318]][a, z]
Out[9]=	4 6 8 10 5 7 9 4 6 8 10 2 a 5 a 4 a a 5 a 9 a 5 a 10 a + 20 a + 13 a + 2 a - ---- - ---- - ---- - --- + ---- + ---- + ---- + 2 2 2 2 z z z z z z z 11 a 5 7 9 11 4 2 6 2 > --- - 16 a z - 27 a z - 13 a z - 2 a z - 14 a z - 37 a z - z 8 2 10 2 12 2 5 3 7 3 9 3 11 3 > 27 a z - 3 a z + a z + 13 a z + 33 a z + 27 a z + 6 a z - 13 3 4 4 6 4 8 4 10 4 12 4 5 5 > a z + 6 a z + 33 a z + 35 a z + 2 a z - 6 a z - 6 a z - 7 5 9 5 11 5 13 5 6 6 8 6 10 6 > 20 a z - 28 a z - 13 a z + a z - 15 a z - 27 a z - 8 a z + 12 6 5 7 7 7 9 7 11 7 6 8 8 8 > 4 a z + 3 a z + 5 a z + 9 a z + 7 a z + 5 a z + 11 a z + 10 8 7 9 9 9 > 6 a z + 2 a z + 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 3 1 3 1 5 3 6 6 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 23 9 21 8 19 8 19 7 17 7 17 6 15 6 q q q t q t q t q t q t q t q t 9 5 6 9 6 6 5 7 1 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 15 5 13 5 13 4 11 4 11 3 9 3 9 2 7 2 7 q t q t q t q t q t q t q t q t q t 4 > ---- 5 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n318

L11n317 L11n319