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

K11a260 K11a262

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X8394 X14,5,15,6 X16,8,17,7 X4,9,5,10 X18,11,19,12 X20,13,21,14 X2,15,3,16 X22,18,1,17 X12,19,13,20 X10,21,11,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 8 14 16 4 18 20 2 22 12 10

Alexander Polynomial:

t-4 - 6t-3 + 16t-2 - 26t-1 + 31 - 26t + 16t2 - 6t3 + t4

Conway Polynomial:

1 + 2z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{129, -4}

Jones Polynomial:

q-10 - 4q-9 + 9q-8 - 14q-7 + 18q-6 - 21q-5 + 20q-4 - 17q-3 + 13q-2 - 7q-1 + 4 - q

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

{...}

A2 (sl(3)) Invariant:

q-30 - q-28 + 2q-24 - 3q-22 + 3q-20 - 2q-18 - q-16 + q-14 - 5q-12 + 4q-10 - 2q-8 + 3q-6 + 3q-4 - q-2 + 2 - q2

HOMFLY-PT Polynomial:

2a2 - a2z2 - 3a2z4 - a2z6 + 6a4z2 + 9a4z4 + 5a4z6 + a4z8 - 2a6 - 7a6z2 - 7a6z4 - 2a6z6 + a8 + 2a8z2 + a8z4

Kauffman Polynomial:

2az3 - 3az5 + az7 - 2a2 - 4a2z2 + 17a2z4 - 15a2z6 + 4a2z8 + 2a3z - a3z3 + 11a3z5 - 15a3z7 + 5a3z9 - 14a4z2 + 44a4z4 - 39a4z6 + 7a4z8 + 2a4z10 + 2a5z + 12a5z5 - 28a5z7 + 12a5z9 + 2a6 - 15a6z2 + 39a6z4 - 48a6z6 + 15a6z8 + 2a6z10 - 2a7z + 14a7z3 - 24a7z5 + a7z7 + 7a7z9 + a8 - 2a8z2 + 3a8z4 - 15a8z6 + 12a8z8 - 2a9z + 10a9z3 - 18a9z5 + 13a9z7 + 3a10z2 - 8a10z4 + 9a10z6 - a11z3 + 4a11z5 + a12z4

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=-4 is the signature of 11261. 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 = 3                       1 j = 1                     3   j = -1                   4 1   j = -3                 9 3     j = -5               9 5       j = -7             11 8         j = -9           10 9           j = -11         8 11             j = -13       6 10               j = -15     3 8                 j = -17   1 6                   j = -19   3                     j = -21 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, 261]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 261]]
Out[3]=	PD[X[6, 2, 7, 1], X[8, 3, 9, 4], X[14, 5, 15, 6], X[16, 8, 17, 7], > X[4, 9, 5, 10], X[18, 11, 19, 12], X[20, 13, 21, 14], X[2, 15, 3, 16], > X[22, 18, 1, 17], X[12, 19, 13, 20], X[10, 21, 11, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 261]]
Out[4]=	GaussCode[1, -8, 2, -5, 3, -1, 4, -2, 5, -11, 6, -10, 7, -3, 8, -4, 9, -6, 10, > -7, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 261]]
Out[5]=	DTCode[6, 8, 14, 16, 4, 18, 20, 2, 22, 12, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 261]][t]
Out[6]=	-4 6 16 26 2 3 4 31 + t - -- + -- - -- - 26 t + 16 t - 6 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 261]][z]
Out[7]=	6 8 1 + 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 261]}
In[9]:=	{KnotDet[Knot[11, Alternating, 261]], KnotSignature[Knot[11, Alternating, 261]]}
Out[9]=	{129, -4}
In[10]:=	J=Jones[Knot[11, Alternating, 261]][q]
Out[10]=	-10 4 9 14 18 21 20 17 13 7 4 + q - -- + -- - -- + -- - -- + -- - -- + -- - - - q 9 8 7 6 5 4 3 2 q q 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, 261]}
In[12]:=	A2Invariant[Knot[11, Alternating, 261]][q]
Out[12]=	-30 -28 2 3 3 2 -16 -14 5 4 2 3 2 + q - q + --- - --- + --- - --- - q + q - --- + --- - -- + -- + 24 22 20 18 12 10 8 6 q q q q q q q q 3 -2 2 > -- - q - q 4 q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 261]][a, z]
Out[13]=	2 6 8 2 2 4 2 6 2 8 2 2 4 4 4 2 a - 2 a + a - a z + 6 a z - 7 a z + 2 a z - 3 a z + 9 a z - 6 4 8 4 2 6 4 6 6 6 4 8 > 7 a z + a z - a z + 5 a z - 2 a z + a z
In[14]:=	Kauffman[Knot[11, Alternating, 261]][a, z]
Out[14]=	2 6 8 3 5 7 9 2 2 4 2 -2 a + 2 a + a + 2 a z + 2 a z - 2 a z - 2 a z - 4 a z - 14 a z - 6 2 8 2 10 2 3 3 3 7 3 9 3 > 15 a z - 2 a z + 3 a z + 2 a z - a z + 14 a z + 10 a z - 11 3 2 4 4 4 6 4 8 4 10 4 12 4 > a z + 17 a z + 44 a z + 39 a z + 3 a z - 8 a z + a z - 5 3 5 5 5 7 5 9 5 11 5 2 6 > 3 a z + 11 a z + 12 a z - 24 a z - 18 a z + 4 a z - 15 a z - 4 6 6 6 8 6 10 6 7 3 7 5 7 > 39 a z - 48 a z - 15 a z + 9 a z + a z - 15 a z - 28 a z + 7 7 9 7 2 8 4 8 6 8 8 8 3 9 > a z + 13 a z + 4 a z + 7 a z + 15 a z + 12 a z + 5 a z + 5 9 7 9 4 10 6 10 > 12 a z + 7 a z + 2 a z + 2 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 261]], Vassiliev[3][Knot[11, Alternating, 261]]}
Out[15]=	{0, 2}
In[16]:=	Kh[Knot[11, Alternating, 261]][q, t]
Out[16]=	5 9 1 3 1 6 3 8 6 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 21 8 19 7 17 7 17 6 15 6 15 5 13 5 q q q t q t q t q t q t q t q t 10 8 11 10 9 11 8 9 3 t > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + --- + 13 4 11 4 11 3 9 3 9 2 7 2 7 5 3 q t q t q t q t q t q t q t q t q 2 4 t t 2 3 3 > --- + -- + 3 q t + q t q q

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a261

K11a260 K11a262