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

K11a321 K11a323

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 12 16 22 20 4 18 8 2 10 14

Alexander Polynomial:

2t-3 - 13t-2 + 36t-1 - 49 + 36t - 13t2 + 2t3

Conway Polynomial:

1 + 2z2 - z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{151, -2}

Jones Polynomial:

- q-8 + 4q-7 - 10q-6 + 16q-5 - 21q-4 + 25q-3 - 24q-2 + 21q-1 - 15 + 9q - 4q2 + q3

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

{K11a147, ...}

A2 (sl(3)) Invariant:

- q-24 + 2q-22 - 3q-20 - 3q-18 + 5q-16 - 3q-14 + 4q-12 + 2q-10 - 2q-8 + 3q-6 - 5q-4 + 4q-2 - 2q2 + 4q4 - 2q6 - q8 + q10

HOMFLY-PT Polynomial:

a-2z2 + 1 - z2 - 2z4 - 2a2 - 2a2z2 + a2z6 + 4a4 + 5a4z2 + 2a4z4 + a4z6 - 2a6 - a6z2 - a6z4

Kauffman Polynomial:

a-2z2 - 2a-2z4 + a-2z6 + 5a-1z3 - 9a-1z5 + 4a-1z7 + 1 - 3z2 + 9z4 - 15z6 + 7z8 + az - 2az3 + 3az5 - 11az7 + 7az9 + 2a2 - 13a2z2 + 29a2z4 - 32a2z6 + 9a2z8 + 3a2z10 + a3z - 8a3z3 + 24a3z5 - 33a3z7 + 16a3z9 + 4a4 - 17a4z2 + 40a4z4 - 42a4z6 + 14a4z8 + 3a4z10 - 3a5z + 8a5z3 - 3a5z5 - 9a5z7 + 9a5z9 + 2a6 - 8a6z2 + 18a6z4 - 22a6z6 + 12a6z8 - 3a7z + 8a7z3 - 14a7z5 + 9a7z7 - 4a8z4 + 4a8z6 - a9z3 + a9z5

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

{2, -4}

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=-2 is the signature of 11322. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                       1 j = 5                     3   j = 3                   6 1   j = 1                 9 3     j = -1               12 6       j = -3             13 10         j = -5           12 11           j = -7         9 13             j = -9       7 12               j = -11     3 9                 j = -13   1 7                   j = -15   3                     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, 322]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 322]]
Out[3]=	PD[X[6, 2, 7, 1], X[12, 4, 13, 3], X[16, 5, 17, 6], X[22, 8, 1, 7], > X[20, 9, 21, 10], X[4, 12, 5, 11], X[18, 13, 19, 14], X[8, 15, 9, 16], > X[2, 17, 3, 18], X[10, 19, 11, 20], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 322]]
Out[4]=	GaussCode[1, -9, 2, -6, 3, -1, 4, -8, 5, -10, 6, -2, 7, -11, 8, -3, 9, -7, 10, > -5, 11, -4]
In[5]:=	DTCode[Knot[11, Alternating, 322]]
Out[5]=	DTCode[6, 12, 16, 22, 20, 4, 18, 8, 2, 10, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 322]][t]
Out[6]=	2 13 36 2 3 -49 + -- - -- + -- + 36 t - 13 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 322]][z]
Out[7]=	2 4 6 1 + 2 z - z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 322]}
In[9]:=	{KnotDet[Knot[11, Alternating, 322]], KnotSignature[Knot[11, Alternating, 322]]}
Out[9]=	{151, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 322]][q]
Out[10]=	-8 4 10 16 21 25 24 21 2 3 -15 - q + -- - -- + -- - -- + -- - -- + -- + 9 q - 4 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, 147], Knot[11, Alternating, 322]}
In[12]:=	A2Invariant[Knot[11, Alternating, 322]][q]
Out[12]=	-24 2 3 3 5 3 4 2 2 3 5 4 2 -q + --- - --- - --- + --- - --- + --- + --- - -- + -- - -- + -- - 2 q + 22 20 18 16 14 12 10 8 6 4 2 q q q q q q q q q q q 4 6 8 10 > 4 q - 2 q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 322]][a, z]
Out[13]=	2 2 4 6 2 z 2 2 4 2 6 2 4 4 4 1 - 2 a + 4 a - 2 a - z + -- - 2 a z + 5 a z - a z - 2 z + 2 a z - 2 a 6 4 2 6 4 6 > a z + a z + a z
In[14]:=	Kauffman[Knot[11, Alternating, 322]][a, z]
Out[14]=	2 2 4 6 3 5 7 2 z 2 2 1 + 2 a + 4 a + 2 a + a z + a z - 3 a z - 3 a z - 3 z + -- - 13 a z - 2 a 3 4 2 6 2 5 z 3 3 3 5 3 7 3 9 3 > 17 a z - 8 a z + ---- - 2 a z - 8 a z + 8 a z + 8 a z - a z + a 4 5 4 2 z 2 4 4 4 6 4 8 4 9 z 5 > 9 z - ---- + 29 a z + 40 a z + 18 a z - 4 a z - ---- + 3 a z + 2 a a 6 3 5 5 5 7 5 9 5 6 z 2 6 4 6 > 24 a z - 3 a z - 14 a z + a z - 15 z + -- - 32 a z - 42 a z - 2 a 7 6 6 8 6 4 z 7 3 7 5 7 7 7 8 > 22 a z + 4 a z + ---- - 11 a z - 33 a z - 9 a z + 9 a z + 7 z + a 2 8 4 8 6 8 9 3 9 5 9 2 10 > 9 a z + 14 a z + 12 a z + 7 a z + 16 a z + 9 a z + 3 a z + 4 10 > 3 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 322]], Vassiliev[3][Knot[11, Alternating, 322]]}
Out[15]=	{2, -4}
In[16]:=	Kh[Knot[11, Alternating, 322]][q, t]
Out[16]=	10 12 1 3 1 7 3 9 7 12 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 9 13 12 11 13 6 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 9 q t + 3 q t + 6 q t + 7 3 7 2 5 2 5 3 q q t q t q t q t q t 3 3 5 3 7 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a322

K11a321 K11a323