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

K11a261 K11a263

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X8394 X14,5,15,6 X16,8,17,7 X4,9,5,10 X20,11,21,12 X18,13,19,14 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, -7, 10, -6, 11, -9}

DT (Dowker-Thistlethwaite) Code:

6 8 14 16 4 20 18 2 22 12 10

Alexander Polynomial:

2t-3 - 11t-2 + 25t-1 - 31 + 25t - 11t2 + 2t3

Conway Polynomial:

1 - z2 + z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a10, ...}

Determinant and Signature:

{107, -2}

Jones Polynomial:

q-9 - 3q-8 + 6q-7 - 10q-6 + 14q-5 - 17q-4 + 17q-3 - 15q-2 + 12q-1 - 7 + 4q - q2

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

{...}

A2 (sl(3)) Invariant:

q-28 - q-24 + 2q-22 - 2q-20 - q-18 + 2q-16 - 3q-14 + 2q-12 - q-10 + 2q-6 - 3q-4 + 4q-2 + 2q4 - q6

HOMFLY-PT Polynomial:

1 - z2 - z4 + a2z2 + 2a2z4 + a2z6 + a4 + 2a4z2 + 2a4z4 + a4z6 - 2a6 - 4a6z2 - 2a6z4 + a8 + a8z2

Kauffman Polynomial:

- a-1z3 + a-1z5 + 1 + 2z2 - 7z4 + 4z6 + az3 - 9az5 + 6az7 - a2z4 - 7a2z6 + 6a2z8 + 2a3z - 8a3z3 + 9a3z5 - 8a3z7 + 5a3z9 + a4 - 12a4z2 + 27a4z4 - 19a4z6 + 4a4z8 + 2a4z10 + 4a5z - 21a5z3 + 43a5z5 - 32a5z7 + 10a5z9 + 2a6 - 18a6z2 + 39a6z4 - 25a6z6 + 3a6z8 + 2a6z10 + a7z - 6a7z3 + 15a7z5 - 15a7z7 + 5a7z9 + a8 - 7a8z2 + 15a8z4 - 16a8z6 + 5a8z8 - a9z + 5a9z3 - 9a9z5 + 3a9z7 + a10z2 - 3a10z4 + a10z6

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

{-1, 3}

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 11262. 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 = 5                       1 j = 3                     3   j = 1                   4 1   j = -1                 8 3     j = -3               8 5       j = -5             9 7         j = -7           8 8           j = -9         6 9             j = -11       4 8               j = -13     2 6                 j = -15   1 4                   j = -17   2                     j = -19 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, 262]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 262]]
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[20, 11, 21, 12], X[18, 13, 19, 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, 262]]
Out[4]=	GaussCode[1, -8, 2, -5, 3, -1, 4, -2, 5, -11, 6, -10, 7, -3, 8, -4, 9, -7, 10, > -6, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 262]]
Out[5]=	DTCode[6, 8, 14, 16, 4, 20, 18, 2, 22, 12, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 262]][t]
Out[6]=	2 11 25 2 3 -31 + -- - -- + -- + 25 t - 11 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 262]][z]
Out[7]=	2 4 6 1 - z + z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 10], Knot[11, Alternating, 262]}
In[9]:=	{KnotDet[Knot[11, Alternating, 262]], KnotSignature[Knot[11, Alternating, 262]]}
Out[9]=	{107, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 262]][q]
Out[10]=	-9 3 6 10 14 17 17 15 12 2 -7 + q - -- + -- - -- + -- - -- + -- - -- + -- + 4 q - q 8 7 6 5 4 3 2 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, 262]}
In[12]:=	A2Invariant[Knot[11, Alternating, 262]][q]
Out[12]=	-28 -24 2 2 -18 2 3 2 -10 2 3 4 4 q - q + --- - --- - q + --- - --- + --- - q + -- - -- + -- + 2 q - 22 20 16 14 12 6 4 2 q q q q q q q q 6 > q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 262]][a, z]
Out[13]=	4 6 8 2 2 2 4 2 6 2 8 2 4 2 4 1 + a - 2 a + a - z + a z + 2 a z - 4 a z + a z - z + 2 a z + 4 4 6 4 2 6 4 6 > 2 a z - 2 a z + a z + a z
In[14]:=	Kauffman[Knot[11, Alternating, 262]][a, z]
Out[14]=	4 6 8 3 5 7 9 2 4 2 1 + a + 2 a + a + 2 a z + 4 a z + a z - a z + 2 z - 12 a z - 3 6 2 8 2 10 2 z 3 3 3 5 3 7 3 > 18 a z - 7 a z + a z - -- + a z - 8 a z - 21 a z - 6 a z + a 5 9 3 4 2 4 4 4 6 4 8 4 10 4 z > 5 a z - 7 z - a z + 27 a z + 39 a z + 15 a z - 3 a z + -- - a 5 3 5 5 5 7 5 9 5 6 2 6 > 9 a z + 9 a z + 43 a z + 15 a z - 9 a z + 4 z - 7 a z - 4 6 6 6 8 6 10 6 7 3 7 5 7 > 19 a z - 25 a z - 16 a z + a z + 6 a z - 8 a z - 32 a z - 7 7 9 7 2 8 4 8 6 8 8 8 3 9 > 15 a z + 3 a z + 6 a z + 4 a z + 3 a z + 5 a z + 5 a z + 5 9 7 9 4 10 6 10 > 10 a z + 5 a z + 2 a z + 2 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 262]], Vassiliev[3][Knot[11, Alternating, 262]]}
Out[15]=	{-1, 3}
In[16]:=	Kh[Knot[11, Alternating, 262]][q, t]
Out[16]=	5 8 1 2 1 4 2 6 4 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q q t q t q t q t q t q t q t 8 6 9 8 8 9 7 8 3 t > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- + --- + 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t q t q t 2 3 2 5 3 > 4 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a262

K11a261 K11a263