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

K11a322 K11a324

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 12 16 22 20 4 18 10 2 14 8

Alexander Polynomial:

2t-3 - 9t-2 + 19t-1 - 23 + 19t - 9t2 + 2t3

Conway Polynomial:

1 + z2 + 3z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{1083, K11a307, ...}

Determinant and Signature:

{83, -2}

Jones Polynomial:

- q-6 + 3q-5 - 6q-4 + 9q-3 - 11q-2 + 13q-1 - 12 + 11q - 8q2 + 5q3 - 3q4 + q5

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

{...}

A2 (sl(3)) Invariant:

- q-18 + q-16 - q-14 - q-12 + 2q-10 - 2q-8 + 2q-6 + q-4 + q-2 + 3 - q2 + q4 - q6 - 2q8 + q10 - q12 + q16

HOMFLY-PT Polynomial:

a-4 + a-4z2 - 3a-2 - 5a-2z2 - 2a-2z4 + 3 + 4z2 + 3z4 + z6 + a2 + 3a2z2 + 3a2z4 + a2z6 - a4 - 2a4z2 - a4z4

Kauffman Polynomial:

a-4 - 5a-4z2 + 8a-4z4 - 5a-4z6 + a-4z8 + a-3z - 15a-3z3 + 27a-3z5 - 16a-3z7 + 3a-3z9 + 3a-2 - 18a-2z2 + 29a-2z4 - 9a-2z6 - 5a-2z8 + 2a-2z10 + a-1z - 20a-1z3 + 52a-1z5 - 40a-1z7 + 9a-1z9 + 3 - 18z2 + 45z4 - 34z6 + 3z8 + 2z10 - az + 6az3 - 14az7 + 6az9 - a2 + 8a2z4 - 21a2z6 + 9a2z8 + 6a3z3 - 19a3z5 + 10a3z7 - a4 + 5a4z2 - 13a4z4 + 9a4z6 + a5z - 4a5z3 + 6a5z5 + 3a6z4 + a7z3

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

{1, -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=-2 is the signature of 11323. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 11                       1 j = 9                     2   j = 7                   3 1   j = 5                 5 2     j = 3               6 3       j = 1             6 5         j = -1           7 6           j = -3         5 7             j = -5       4 6               j = -7     2 5                 j = -9   1 4                   j = -11   2                     j = -13 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, 323]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 323]]
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, 10, 21, 9], X[4, 12, 5, 11], X[18, 13, 19, 14], X[10, 15, 11, 16], > X[2, 17, 3, 18], X[14, 19, 15, 20], X[8, 22, 9, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 323]]
Out[4]=	GaussCode[1, -9, 2, -6, 3, -1, 4, -11, 5, -8, 6, -2, 7, -10, 8, -3, 9, -7, 10, > -5, 11, -4]
In[5]:=	DTCode[Knot[11, Alternating, 323]]
Out[5]=	DTCode[6, 12, 16, 22, 20, 4, 18, 10, 2, 14, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 323]][t]
Out[6]=	2 9 19 2 3 -23 + -- - -- + -- + 19 t - 9 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 323]][z]
Out[7]=	2 4 6 1 + z + 3 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 83], Knot[11, Alternating, 307], Knot[11, Alternating, 323]}
In[9]:=	{KnotDet[Knot[11, Alternating, 323]], KnotSignature[Knot[11, Alternating, 323]]}
Out[9]=	{83, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 323]][q]
Out[10]=	-6 3 6 9 11 13 2 3 4 5 -12 - q + -- - -- + -- - -- + -- + 11 q - 8 q + 5 q - 3 q + q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 323]}
In[12]:=	A2Invariant[Knot[11, Alternating, 323]][q]
Out[12]=	-18 -16 -14 -12 2 2 2 -4 -2 2 4 6 3 - q + q - q - q + --- - -- + -- + q + q - q + q - q - 10 8 6 q q q 8 10 12 16 > 2 q + q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 323]][a, z]
Out[13]=	2 2 4 -4 3 2 4 2 z 5 z 2 2 4 2 4 2 z 3 + a - -- + a - a + 4 z + -- - ---- + 3 a z - 2 a z + 3 z - ---- + 2 4 2 2 a a a a 2 4 4 4 6 2 6 > 3 a z - a z + z + a z
In[14]:=	Kauffman[Knot[11, Alternating, 323]][a, z]
Out[14]=	2 2 -4 3 2 4 z z 5 2 5 z 18 z 4 2 3 + a + -- - a - a + -- + - - a z + a z - 18 z - ---- - ----- + 5 a z - 2 3 a 4 2 a a a a 3 3 4 4 15 z 20 z 3 3 3 5 3 7 3 4 8 z 29 z > ----- - ----- + 6 a z + 6 a z - 4 a z + a z + 45 z + ---- + ----- + 3 a 4 2 a a a 5 5 2 4 4 4 6 4 27 z 52 z 3 5 5 5 6 > 8 a z - 13 a z + 3 a z + ----- + ----- - 19 a z + 6 a z - 34 z - 3 a a 6 6 7 7 5 z 9 z 2 6 4 6 16 z 40 z 7 3 7 > ---- - ---- - 21 a z + 9 a z - ----- - ----- - 14 a z + 10 a z + 4 2 3 a a a a 8 8 9 9 10 8 z 5 z 2 8 3 z 9 z 9 10 2 z > 3 z + -- - ---- + 9 a z + ---- + ---- + 6 a z + 2 z + ----- 4 2 3 a 2 a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 323]], Vassiliev[3][Knot[11, Alternating, 323]]}
Out[15]=	{1, -2}
In[16]:=	Kh[Knot[11, Alternating, 323]][q, t]
Out[16]=	7 7 1 2 1 4 2 5 4 6 -- + - + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 3 q 13 5 11 4 9 4 9 3 7 3 7 2 5 2 5 q q t q t q t q t q t q t q t q t 5 6 t 2 3 2 3 3 5 3 5 4 > ---- + --- + 6 q t + 5 q t + 6 q t + 3 q t + 5 q t + 2 q t + 3 q q t 7 4 7 5 9 5 11 6 > 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a323

K11a322 K11a324