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

K11a83 K11a85

Knotscape

This page is passe. Go here instead!    The Knot K11a84 Visit K11a84's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X4251 X10,4,11,3 X12,5,13,6 X16,8,17,7 X2,10,3,9 X22,11,1,12 X20,13,21,14 X18,15,19,16 X6,18,7,17 X14,19,15,20 X8,21,9,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 12 16 2 22 20 18 6 14 8

Alexander Polynomial:

- 2t-3 + 10t-2 - 23t-1 + 31 - 23t + 10t2 - 2t3

Conway Polynomial:

1 - z2 - 2z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{10119, ...}

Determinant and Signature:

{101, 0}

Jones Polynomial:

q-6 - 3q-5 + 6q-4 - 10q-3 + 14q-2 - 16q-1 + 16 - 14q + 11q2 - 6q3 + 3q4 - q5

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

{K11a88, ...}

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

- a-4 - a-4z2 + 3a-2 + 4a-2z2 + 2a-2z4 - 1 - 2z2 - 2z4 - z6 - a2 - 4a2z2 - 3a2z4 - a2z6 + a4 + 2a4z2 + a4z4

Kauffman Polynomial:

a-5z - 2a-5z3 + a-5z5 - a-4 + 4a-4z2 - 6a-4z4 + 3a-4z6 + 2a-3z - 2a-3z3 - 4a-3z5 + 4a-3z7 - 3a-2 + 9a-2z2 - 8a-2z4 - a-2z6 + 4a-2z8 + 2a-1z - 3a-1z3 - a-1z5 + 3a-1z9 - 1 + z2 + 6z4 - 11z6 + 6z8 + z10 + 2az - 7az3 + 9az5 - 10az7 + 6az9 + a2 - 9a2z2 + 18a2z4 - 18a2z6 + 6a2z8 + a2z10 + 3a3z3 - 4a3z5 - 3a3z7 + 3a3z9 + a4 - 3a4z2 + 7a4z4 - 10a4z6 + 4a4z8 - a5z + 7a5z3 - 9a5z5 + 3a5z7 + 2a6z2 - 3a6z4 + a6z6

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=0 is the signature of 1184. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 11                       1 j = 9                     2   j = 7                   4 1   j = 5                 7 2     j = 3               7 4       j = 1             9 7         j = -1           8 8           j = -3         6 8             j = -5       4 8               j = -7     2 6                 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, 84]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 84]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[12, 5, 13, 6], X[16, 8, 17, 7], > X[2, 10, 3, 9], X[22, 11, 1, 12], X[20, 13, 21, 14], X[18, 15, 19, 16], > X[6, 18, 7, 17], X[14, 19, 15, 20], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 84]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -9, 4, -11, 5, -2, 6, -3, 7, -10, 8, -4, 9, -8, 10, > -7, 11, -6]
In[5]:=	DTCode[Knot[11, Alternating, 84]]
Out[5]=	DTCode[4, 10, 12, 16, 2, 22, 20, 18, 6, 14, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 84]][t]
Out[6]=	2 10 23 2 3 31 - -- + -- - -- - 23 t + 10 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 84]][z]
Out[7]=	2 4 6 1 - z - 2 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 119], Knot[11, Alternating, 84]}
In[9]:=	{KnotDet[Knot[11, Alternating, 84]], KnotSignature[Knot[11, Alternating, 84]]}
Out[9]=	{101, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 84]][q]
Out[10]=	-6 3 6 10 14 16 2 3 4 5 16 + q - -- + -- - -- + -- - -- - 14 q + 11 q - 6 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, 84], Knot[11, Alternating, 88]}
In[12]:=	A2Invariant[Knot[11, Alternating, 84]][q]
Out[12]=	-18 -16 -14 -12 3 3 -6 -4 -2 2 4 -3 + q - q + q + q - --- + -- - q - q + q + 3 q - q + 10 8 q q 6 8 10 12 16 > 2 q + 3 q - 2 q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 84]][a, z]
Out[13]=	2 2 4 -4 3 2 4 2 z 4 z 2 2 4 2 4 2 z -1 - a + -- - a + a - 2 z - -- + ---- - 4 a z + 2 a z - 2 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, 84]][a, z]
Out[14]=	2 2 -4 3 2 4 z 2 z 2 z 5 2 4 z 9 z -1 - a - -- + a + a + -- + --- + --- + 2 a z - a z + z + ---- + ---- - 2 5 3 a 4 2 a a a a a 3 3 3 2 2 4 2 6 2 2 z 2 z 3 z 3 3 3 > 9 a z - 3 a z + 2 a z - ---- - ---- - ---- - 7 a z + 3 a z + 5 3 a a a 4 4 5 5 5 3 4 6 z 8 z 2 4 4 4 6 4 z 4 z > 7 a z + 6 z - ---- - ---- + 18 a z + 7 a z - 3 a z + -- - ---- - 4 2 5 3 a a a a 5 6 6 z 5 3 5 5 5 6 3 z z 2 6 4 6 > -- + 9 a z - 4 a z - 9 a z - 11 z + ---- - -- - 18 a z - 10 a z + a 4 2 a a 7 8 6 6 4 z 7 3 7 5 7 8 4 z 2 8 > a z + ---- - 10 a z - 3 a z + 3 a z + 6 z + ---- + 6 a z + 3 2 a a 9 4 8 3 z 9 3 9 10 2 10 > 4 a z + ---- + 6 a z + 3 a z + z + a z a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 84]], Vassiliev[3][Knot[11, Alternating, 84]]}
Out[15]=	{-1, 2}
In[16]:=	Kh[Knot[11, Alternating, 84]][q, t]
Out[16]=	8 1 2 1 4 2 6 4 8 - + 9 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 6 8 8 3 3 2 5 2 5 3 > ----- + ---- + --- + 7 q t + 7 q t + 4 q t + 7 q t + 2 q t + 3 2 3 q t q t q t 7 3 7 4 9 4 11 5 > 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a84

K11a83 K11a85