© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11a332 K11a334

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X14,3,15,4 X12,5,13,6 X16,8,17,7 X20,9,21,10 X18,11,19,12 X4,13,5,14 X2,15,3,16 X22,18,1,17 X10,19,11,20 X8,21,9,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 14 12 16 20 18 4 2 22 10 8

Alexander Polynomial:

4t-2 - 16t-1 + 25 - 16t + 4t2

Conway Polynomial:

1 + 4z4

Other knots with the same Alexander/Conway Polynomial:

{1033, ...}

Determinant and Signature:

{65, 0}

Jones Polynomial:

q-8 - 2q-7 + 3q-6 - 6q-5 + 8q-4 - 9q-3 + 10q-2 - 9q-1 + 8 - 5q + 3q2 - q3

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

q-26 + q-24 - q-22 - q-18 - 3q-16 + q-14 + 2q-8 + 2q-4 + 2q2 - 2q4 + q6 + q8 - q10

HOMFLY-PT Polynomial:

- a-2z2 + z4 + 2a2 + 3a2z2 + 2a2z4 + a4z4 - 2a6 - 2a6z2 + a8

Kauffman Polynomial:

a-3z3 - a-2z2 + 3a-2z4 - 3a-1z3 + 5a-1z5 + 4z2 - 10z4 + 7z6 + 5az3 - 13az5 + 7az7 - 2a2 + 7a2z2 - 5a2z4 - 8a2z6 + 5a2z8 + 4a3z - a3z3 - 3a3z5 - 5a3z7 + 3a3z9 - 6a4z2 + 19a4z4 - 17a4z6 + 2a4z8 + a4z10 + 8a5z - 28a5z3 + 39a5z5 - 24a5z7 + 5a5z9 + 2a6 - 14a6z2 + 22a6z4 - 8a6z6 - 2a6z8 + a6z10 + 4a7z - 18a7z3 + 24a7z5 - 12a7z7 + 2a7z9 + a8 - 6a8z2 + 11a8z4 - 6a8z6 + a8z8

V2 and V3, the type 2 and 3 Vassiliev invariants:

{0, 2}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=0 is the signature of 11333. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 7                       1 j = 5                     2   j = 3                   3 1   j = 1                 5 2     j = -1               5 4       j = -3             5 4         j = -5           4 5           j = -7         4 5             j = -9       2 4               j = -11     1 4                 j = -13   1 2                   j = -15   1                     j = -17 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]:=	Show[DrawMorseLink[Knot[11, Alternating, 333]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 333]]
Out[3]=	PD[X[6, 2, 7, 1], X[14, 3, 15, 4], X[12, 5, 13, 6], X[16, 8, 17, 7], > X[20, 9, 21, 10], X[18, 11, 19, 12], X[4, 13, 5, 14], X[2, 15, 3, 16], > X[22, 18, 1, 17], X[10, 19, 11, 20], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 333]]
Out[4]=	GaussCode[1, -8, 2, -7, 3, -1, 4, -11, 5, -10, 6, -3, 7, -2, 8, -4, 9, -6, 10, > -5, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 333]]
Out[5]=	DTCode[6, 14, 12, 16, 20, 18, 4, 2, 22, 10, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 333]][t]
Out[6]=	4 16 2 25 + -- - -- - 16 t + 4 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 333]][z]
Out[7]=	4 1 + 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 33], Knot[11, Alternating, 333]}
In[9]:=	{KnotDet[Knot[11, Alternating, 333]], KnotSignature[Knot[11, Alternating, 333]]}
Out[9]=	{65, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 333]][q]
Out[10]=	-8 2 3 6 8 9 10 9 2 3 8 + q - -- + -- - -- + -- - -- + -- - - - 5 q + 3 q - q 7 6 5 4 3 2 q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 333]}
In[12]:=	A2Invariant[Knot[11, Alternating, 333]][q]
Out[12]=	-26 -24 -22 -18 3 -14 2 2 2 4 6 8 10 q + q - q - q - --- + q + -- + -- + 2 q - 2 q + q + q - q 16 8 4 q q q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 333]][a, z]
Out[13]=	2 2 6 8 z 2 2 6 2 4 2 4 4 4 2 a - 2 a + a - -- + 3 a z - 2 a z + z + 2 a z + a z 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 333]][a, z]
Out[14]=	2 2 6 8 3 5 7 2 z 2 2 4 2 -2 a + 2 a + a + 4 a z + 8 a z + 4 a z + 4 z - -- + 7 a z - 6 a z - 2 a 3 3 6 2 8 2 z 3 z 3 3 3 5 3 7 3 > 14 a z - 6 a z + -- - ---- + 5 a z - a z - 28 a z - 18 a z - 3 a a 4 5 4 3 z 2 4 4 4 6 4 8 4 5 z 5 > 10 z + ---- - 5 a z + 19 a z + 22 a z + 11 a z + ---- - 13 a z - 2 a a 3 5 5 5 7 5 6 2 6 4 6 6 6 > 3 a z + 39 a z + 24 a z + 7 z - 8 a z - 17 a z - 8 a z - 8 6 7 3 7 5 7 7 7 2 8 4 8 > 6 a z + 7 a z - 5 a z - 24 a z - 12 a z + 5 a z + 2 a z - 6 8 8 8 3 9 5 9 7 9 4 10 6 10 > 2 a z + a z + 3 a z + 5 a z + 2 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 333]], Vassiliev[3][Knot[11, Alternating, 333]]}
Out[15]=	{0, 2}
In[16]:=	Kh[Knot[11, Alternating, 333]][q, t]
Out[16]=	4 1 1 1 2 1 4 2 4 - + 5 q + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + q 17 8 15 7 13 7 13 6 11 6 11 5 9 5 9 4 q t q t q t q t q t q t q t q t 4 5 4 5 5 4 5 3 > ----- + ----- + ----- + ----- + ----- + ---- + --- + 2 q t + 3 q t + 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t 3 2 5 2 7 3 > q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a333

K11a332 K11a334