© | Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table:

L11a542 L11a544

Knotscape

This page is passe. Go here instead! The 4-Component Link L11a543 Visit L11a543's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

q-27/2 - 4q-25/2 + 8q-23/2 - 14q-21/2 + 17q-19/2 - 22q-17/2 + 19q-15/2 - 19q-13/2 + 12q-11/2 - 8q-9/2 + 3q-7/2 - q-5/2

A2 (sl(3)) Invariant:

- q-42 + 4q-38 + q-36 + 6q-34 + 10q-32 + 6q-30 + 14q-28 + 10q-26 + 11q-24 + 10q-22 + 2q-20 + 7q-18 - q-16 + 3q-12 - 2q-10 + q-8

HOMFLY-PT Polynomial:

- a5z - 2a5z3 - a5z5 - a7z-3 - 5a7z-1 - 8a7z - 8a7z3 - 3a7z5 + 3a9z-3 + 10a9z-1 + 8a9z - a9z3 - 2a9z5 - 3a11z-3 - 5a11z-1 + 2a11z + 3a11z3 + a13z-3 - a13z

Kauffman Polynomial:

- a5z + 2a5z3 - a5z5 - a6z2 + 4a6z4 - 3a6z6 + a7z-3 - 5a7z-1 + 11a7z - 13a7z3 + 11a7z5 - 6a7z7 - 3a8z-2 + 10a8 - 15a8z2 + 9a8z4 + 3a8z6 - 6a8z8 + 3a9z-3 - 12a9z-1 + 29a9z - 39a9z3 + 29a9z5 - 7a9z7 - 4a9z9 - 6a10z-2 + 19a10 - 18a10z2 - 12a10z4 + 31a10z6 - 15a10z8 - a10z10 + 3a11z-3 - 12a11z-1 + 29a11z - 49a11z3 + 38a11z5 - a11z7 - 8a11z9 - 3a12z-2 + 10a12 - 4a12z2 - 23a12z4 + 38a12z6 - 15a12z8 - a12z10 + a13z-3 - 5a13z-1 + 15a13z - 33a13z3 + 31a13z5 - 4a13z7 - 4a13z9 - a14z2 - 4a14z4 + 12a14z6 - 6a14z8 + 3a15z - 8a15z3 + 10a15z5 - 4a15z7 - a16z2 + 2a16z4 - a16z6

Khovanov Homology:

trqj r = -11 r = -10 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                   5     j = -10                 7 3     j = -12               12 5       j = -14             9 9         j = -16           13 10           j = -18         8 13             j = -20       6 9               j = -22     3 9                 j = -24   1 5                   j = -26   3                     j = -28 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, Alternating, 543]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 543]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[10, 13, 5, 14], X[8, 17, 9, 18], > X[14, 7, 15, 8], X[18, 9, 19, 10], X[22, 19, 17, 20], X[16, 21, 11, 22], > X[20, 15, 21, 16], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 5, -4, 6, -3}, {11, -2, 3, -5, 9, -8}, > {4, -6, 7, -9, 8, -7}]
In[6]:=	Jones[L][q]
Out[6]=	-(27/2) 4 8 14 17 22 19 19 12 q - ----- + ----- - ----- + ----- - ----- + ----- - ----- + ----- - 25/2 23/2 21/2 19/2 17/2 15/2 13/2 11/2 q q q q q q q q 8 3 -(5/2) > ---- + ---- - q 9/2 7/2 q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-42 4 -36 6 10 6 14 10 11 10 2 7 -q + --- + q + --- + --- + --- + --- + --- + --- + --- + --- + --- - 38 34 32 30 28 26 24 22 20 18 q q q q q q q q q q -16 3 2 -8 > q + --- - --- + q 12 10 q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 543]][a, z]
Out[8]=	7 9 11 13 7 9 11 a 3 a 3 a a 5 a 10 a 5 a 5 7 9 -(--) + ---- - ----- + --- - ---- + ----- - ----- - a z - 8 a z + 8 a z + 3 3 3 3 z z z z z z z 11 13 5 3 7 3 9 3 11 3 5 5 7 5 > 2 a z - a z - 2 a z - 8 a z - a z + 3 a z - a z - 3 a z - 9 5 > 2 a z
In[9]:=	Kauffman[Link[11, Alternating, 543]][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 5 7 9 11 > ---- - ----- - ------ - ----- - a z + 11 a z + 29 a z + 29 a z + z z z z 13 15 6 2 8 2 10 2 12 2 14 2 > 15 a z + 3 a z - a z - 15 a z - 18 a z - 4 a z - a z - 16 2 5 3 7 3 9 3 11 3 13 3 15 3 > a z + 2 a z - 13 a z - 39 a z - 49 a z - 33 a z - 8 a z + 6 4 8 4 10 4 12 4 14 4 16 4 5 5 > 4 a z + 9 a z - 12 a z - 23 a z - 4 a z + 2 a z - a z + 7 5 9 5 11 5 13 5 15 5 6 6 > 11 a z + 29 a z + 38 a z + 31 a z + 10 a z - 3 a z + 8 6 10 6 12 6 14 6 16 6 7 7 9 7 > 3 a z + 31 a z + 38 a z + 12 a z - a z - 6 a z - 7 a z - 11 7 13 7 15 7 8 8 10 8 12 8 14 8 > a z - 4 a z - 4 a z - 6 a z - 15 a z - 15 a z - 6 a z - 9 9 11 9 13 9 10 10 12 10 > 4 a z - 8 a z - 4 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	-6 -4 1 3 1 5 3 9 6 q + q + ------- + ------- + ------- + ------ + ------ + ------ + ------ + 28 11 26 10 24 10 24 9 22 9 22 8 20 8 q t q t q t q t q t q t q t 9 8 13 13 10 9 9 12 > ------ + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 20 7 18 7 18 6 16 6 16 5 14 5 14 4 12 4 q t q t q t q t q t q t q t q t 5 7 3 5 3 > ------ + ------ + ------ + ----- + ---- 12 3 10 3 10 2 8 2 6 q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a543

L11a542 L11a544