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

K11a293 K11a295

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 10 16 22 4 18 20 8 2 14 12

Alexander Polynomial:

2t-3 - 11t-2 + 29t-1 - 39 + 29t - 11t2 + 2t3

Conway Polynomial:

1 + 3z2 + z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a178, ...}

Determinant and Signature:

{123, -2}

Jones Polynomial:

- q-8 + 3q-7 - 7q-6 + 12q-5 - 17q-4 + 20q-3 - 19q-2 + 18q-1 - 13 + 8q - 4q2 + q3

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

{K11a146, ...}

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

a-2z2 - 2z2 - 2z4 + a2z4 + a2z6 + 3a4 + 6a4z2 + 3a4z4 + a4z6 - 2a6 - 2a6z2 - a6z4

Kauffman Polynomial:

a-2z2 - 2a-2z4 + a-2z6 + a-1z + 6a-1z3 - 10a-1z5 + 4a-1z7 - z2 + 6z4 - 13z6 + 6z8 + 3az - 4az3 - 2az5 - 6az7 + 5az9 - 10a2z2 + 19a2z4 - 23a2z6 + 7a2z8 + 2a2z10 + 2a3z - 12a3z3 + 24a3z5 - 25a3z7 + 11a3z9 + 3a4 - 18a4z2 + 40a4z4 - 32a4z6 + 9a4z8 + 2a4z10 - 4a5z + 11a5z3 + 2a5z5 - 9a5z7 + 6a5z9 + 2a6 - 10a6z2 + 24a6z4 - 20a6z6 + 8a6z8 - 4a7z + 11a7z3 - 13a7z5 + 6a7z7 - 5a8z4 + 3a8z6 - 2a9z3 + a9z5

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

{3, -5}

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 11294. 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                   5 1   j = 1                 8 3     j = -1               10 5       j = -3             10 9         j = -5           10 9           j = -7         7 10             j = -9       5 10               j = -11     2 7                 j = -13   1 5                   j = -15   2                     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, 294]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 294]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 4, 11, 3], X[16, 5, 17, 6], X[22, 8, 1, 7], > X[4, 10, 5, 9], X[18, 11, 19, 12], X[20, 13, 21, 14], X[8, 15, 9, 16], > X[2, 17, 3, 18], X[14, 19, 15, 20], X[12, 21, 13, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 294]]
Out[4]=	GaussCode[1, -9, 2, -5, 3, -1, 4, -8, 5, -2, 6, -11, 7, -10, 8, -3, 9, -6, 10, > -7, 11, -4]
In[5]:=	DTCode[Knot[11, Alternating, 294]]
Out[5]=	DTCode[6, 10, 16, 22, 4, 18, 20, 8, 2, 14, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 294]][t]
Out[6]=	2 11 29 2 3 -39 + -- - -- + -- + 29 t - 11 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 294]][z]
Out[7]=	2 4 6 1 + 3 z + z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 178], Knot[11, Alternating, 294]}
In[9]:=	{KnotDet[Knot[11, Alternating, 294]], KnotSignature[Knot[11, Alternating, 294]]}
Out[9]=	{123, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 294]][q]
Out[10]=	-8 3 7 12 17 20 19 18 2 3 -13 - q + -- - -- + -- - -- + -- - -- + -- + 8 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, 146], Knot[11, Alternating, 294]}
In[12]:=	A2Invariant[Knot[11, Alternating, 294]][q]
Out[12]=	-24 -22 2 2 4 3 2 2 -8 4 3 3 2 -q + q - --- - --- + --- - --- + --- + --- - q + -- - -- + -- - 2 q + 20 18 16 14 12 10 6 4 2 q q q q q q q q q 4 6 8 10 > 3 q - 2 q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 294]][a, z]
Out[13]=	2 4 6 2 z 4 2 6 2 4 2 4 4 4 6 4 3 a - 2 a - 2 z + -- + 6 a z - 2 a z - 2 z + a z + 3 a z - a z + 2 a 2 6 4 6 > a z + a z
In[14]:=	Kauffman[Knot[11, Alternating, 294]][a, z]
Out[14]=	2 4 6 z 3 5 7 2 z 2 2 3 a + 2 a + - + 3 a z + 2 a z - 4 a z - 4 a z - z + -- - 10 a z - a 2 a 3 4 2 6 2 6 z 3 3 3 5 3 7 3 > 18 a z - 10 a z + ---- - 4 a z - 12 a z + 11 a z + 11 a z - a 4 5 9 3 4 2 z 2 4 4 4 6 4 8 4 10 z > 2 a z + 6 z - ---- + 19 a z + 40 a z + 24 a z - 5 a z - ----- - 2 a a 6 5 3 5 5 5 7 5 9 5 6 z 2 6 > 2 a z + 24 a z + 2 a z - 13 a z + a z - 13 z + -- - 23 a z - 2 a 7 4 6 6 6 8 6 4 z 7 3 7 5 7 > 32 a z - 20 a z + 3 a z + ---- - 6 a z - 25 a z - 9 a z + a 7 7 8 2 8 4 8 6 8 9 3 9 > 6 a z + 6 z + 7 a z + 9 a z + 8 a z + 5 a z + 11 a z + 5 9 2 10 4 10 > 6 a z + 2 a z + 2 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 294]], Vassiliev[3][Knot[11, Alternating, 294]]}
Out[15]=	{3, -5}
In[16]:=	Kh[Knot[11, Alternating, 294]][q, t]
Out[16]=	9 10 1 2 1 5 2 7 5 10 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 7 10 10 9 10 5 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 8 q t + 3 q t + 5 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 K11a294

K11a293 K11a295