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L11n305 L11n307

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DrawMorseLink

PD Presentation:

X6172 X3,13,4,12 X15,22,16,11 X13,20,14,21 X21,14,22,15 X17,8,18,9 X7,16,8,17 X9,18,10,19 X19,10,20,5 X2536 X11,1,12,4

Gauss Code:

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

Jones Polynomial:

- q-10 + 2q-9 - 3q-8 + 4q-7 - 4q-6 + 5q-5 - 3q-4 + 4q-3 - q-2 + q-1

A2 (sl(3)) Invariant:

- q-34 - 2q-30 - q-28 - q-26 + 2q-22 + q-20 + 5q-18 + 4q-16 + 6q-14 + 5q-12 + 4q-10 + 3q-8 + q-6 + q-4

HOMFLY-PT Polynomial:

2a4z-2 + 9a4 + 12a4z2 + 6a4z4 + a4z6 - 5a6z-2 - 18a6 - 24a6z2 - 18a6z4 - 7a6z6 - a6z8 + 4a8z-2 + 11a8 + 12a8z2 + 6a8z4 + a8z6 - a10z-2 - 2a10 - a10z2

Kauffman Polynomial:

- 2a4z-2 + 11a4 - 21a4z2 + 18a4z4 - 7a4z6 + a4z8 + 5a5z-1 - 19a5z + 18a5z3 - a5z5 - 4a5z7 + a5z9 - 5a6z-2 + 22a6 - 46a6z2 + 55a6z4 - 29a6z6 + 5a6z8 + 9a7z-1 - 35a7z + 50a7z3 - 25a7z5 + a7z7 + a7z9 - 4a8z-2 + 13a8 - 21a8z2 + 27a8z4 - 19a8z6 + 4a8z8 + 5a9z-1 - 19a9z + 31a9z3 - 23a9z5 + 5a9z7 - a10z-2 + 6a10z2 - 10a10z4 + 3a10z6 + a11z-1 - 2a11z - a11z3 + a11z5 - a12 + 2a12z2 + a13z

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = -1                   1 j = -3                     j = -5               4 1   j = -7             1 2     j = -9           4 2       j = -11       1 2 2         j = -13       3 3           j = -15   1 2 2             j = -17   2 3               j = -19   1                 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, NonAlternating, 306]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 306]]
Out[4]=	PD[X[6, 1, 7, 2], X[3, 13, 4, 12], X[15, 22, 16, 11], X[13, 20, 14, 21], > X[21, 14, 22, 15], X[17, 8, 18, 9], X[7, 16, 8, 17], X[9, 18, 10, 19], > X[19, 10, 20, 5], X[2, 5, 3, 6], X[11, 1, 12, 4]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, -2, 11}, {10, -1, -7, 6, -8, 9}, > {-11, 2, -4, 5, -3, 7, -6, 8, -9, 4, -5, 3}]
In[6]:=	Jones[L][q]
Out[6]=	-10 2 3 4 4 5 3 4 -2 1 -q + -- - -- + -- - -- + -- - -- + -- - q + - 9 8 7 6 5 4 3 q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-34 2 -28 -26 2 -20 5 4 6 5 4 3 -q - --- - q - q + --- + q + --- + --- + --- + --- + --- + -- + 30 22 18 16 14 12 10 8 q q q q q q q q -6 -4 > q + q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 306]][a, z]
Out[8]=	4 6 8 10 4 6 8 10 2 a 5 a 4 a a 4 2 6 2 9 a - 18 a + 11 a - 2 a + ---- - ---- + ---- - --- + 12 a z - 24 a z + 2 2 2 2 z z z z 8 2 10 2 4 4 6 4 8 4 4 6 6 6 > 12 a z - a z + 6 a z - 18 a z + 6 a z + a z - 7 a z + 8 6 6 8 > a z - a z
In[9]:=	Kauffman[Link[11, NonAlternating, 306]][a, z]
Out[9]=	4 6 8 10 5 7 9 4 6 8 12 2 a 5 a 4 a a 5 a 9 a 5 a 11 a + 22 a + 13 a - a - ---- - ---- - ---- - --- + ---- + ---- + ---- + 2 2 2 2 z z z z z z z 11 a 5 7 9 11 13 4 2 6 2 > --- - 19 a z - 35 a z - 19 a z - 2 a z + a z - 21 a z - 46 a z - z 8 2 10 2 12 2 5 3 7 3 9 3 11 3 > 21 a z + 6 a z + 2 a z + 18 a z + 50 a z + 31 a z - a z + 4 4 6 4 8 4 10 4 5 5 7 5 9 5 > 18 a z + 55 a z + 27 a z - 10 a z - a z - 25 a z - 23 a z + 11 5 4 6 6 6 8 6 10 6 5 7 7 7 > a z - 7 a z - 29 a z - 19 a z + 3 a z - 4 a z + a z + 9 7 4 8 6 8 8 8 5 9 7 9 > 5 a z + a z + 5 a z + 4 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	2 4 1 1 2 1 3 2 2 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 7 5 21 7 19 6 17 6 15 6 17 5 15 5 15 4 q q q t q t q t q t q t q t q t 2 3 1 3 2 2 4 2 1 t t > ------ + ------ + ------ + ------ + ------ + ----- + ---- + ---- + -- + -- 13 4 11 4 13 3 11 3 11 2 9 2 9 7 5 q q t q t q t q t q t q t q t q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n306

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