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

K11a207 K11a209

Knotscape

This page is passe. Go here instead!    The Knot K11a208 Visit K11a208's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 16 18 20 14 2 22 10 8 6

Alexander Polynomial:

- 2t-3 + 12t-2 - 24t-1 + 29 - 24t + 12t2 - 2t3

Conway Polynomial:

1 + 6z2 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{105, -4}

Jones Polynomial:

- q-11 + 3q-10 - 7q-9 + 11q-8 - 15q-7 + 17q-6 - 16q-5 + 15q-4 - 10q-3 + 6q-2 - 3q-1 + 1

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

{...}

A2 (sl(3)) Invariant:

- q-34 + q-30 - 3q-28 + q-26 - 2q-22 + 3q-20 - 2q-18 + 3q-16 + q-14 + 4q-10 - 3q-8 + q-6 - q-2 + 1

HOMFLY-PT Polynomial:

2a2z2 + a2z4 + a4 - 2a4z4 - a4z6 + a6 + 2a6z2 - a6z4 - a6z6 + 3a8z2 + 2a8z4 - a10 - a10z2

Kauffman Polynomial:

2a2z2 - 3a2z4 + a2z6 + 7a3z3 - 9a3z5 + 3a3z7 + a4 - 4a4z2 + 8a4z4 - 10a4z6 + 4a4z8 + a5z - 2a5z5 - 3a5z7 + 3a5z9 - a6 - 4a6z2 + 11a6z4 - 15a6z6 + 6a6z8 + a6z10 + 4a7z - 8a7z3 + 6a7z5 - 8a7z7 + 6a7z9 - a8z2 + 6a8z4 - 12a8z6 + 7a8z8 + a8z10 + 6a9z3 - 10a9z5 + 3a9z7 + 3a9z9 + a10 - a10z2 + a10z4 - 5a10z6 + 5a10z8 - 2a11z + 5a11z3 - 8a11z5 + 5a11z7 + 2a12z2 - 5a12z4 + 3a12z6 + a13z - 2a13z3 + a13z5

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

{6, -15}

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 11208. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 1                       1 j = -1                     2   j = -3                   4 1   j = -5                 7 3     j = -7               8 3       j = -9             8 7         j = -11           9 8           j = -13         6 8             j = -15       5 9               j = -17     2 6                 j = -19   1 5                   j = -21   2                     j = -23 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, 208]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 208]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 3, 13, 4], X[16, 5, 17, 6], X[18, 7, 19, 8], > X[20, 9, 21, 10], X[14, 12, 15, 11], X[2, 13, 3, 14], X[22, 15, 1, 16], > X[10, 17, 11, 18], X[8, 19, 9, 20], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 208]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -11, 4, -10, 5, -9, 6, -2, 7, -6, 8, -3, 9, -4, 10, > -5, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 208]]
Out[5]=	DTCode[4, 12, 16, 18, 20, 14, 2, 22, 10, 8, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 208]][t]
Out[6]=	2 12 24 2 3 29 - -- + -- - -- - 24 t + 12 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 208]][z]
Out[7]=	2 6 1 + 6 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 208]}
In[9]:=	{KnotDet[Knot[11, Alternating, 208]], KnotSignature[Knot[11, Alternating, 208]]}
Out[9]=	{105, -4}
In[10]:=	J=Jones[Knot[11, Alternating, 208]][q]
Out[10]=	-11 3 7 11 15 17 16 15 10 6 3 1 - q + --- - -- + -- - -- + -- - -- + -- - -- + -- - - 10 9 8 7 6 5 4 3 2 q 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, 208]}
In[12]:=	A2Invariant[Knot[11, Alternating, 208]][q]
Out[12]=	-34 -30 3 -26 2 3 2 3 -14 4 3 -6 1 - q + q - --- + q - --- + --- - --- + --- + q + --- - -- + q - 28 22 20 18 16 10 8 q q q q q q q -2 > q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 208]][a, z]
Out[13]=	4 6 10 2 2 6 2 8 2 10 2 2 4 4 4 a + a - a + 2 a z + 2 a z + 3 a z - a z + a z - 2 a z - 6 4 8 4 4 6 6 6 > a z + 2 a z - a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 208]][a, z]
Out[14]=	4 6 10 5 7 11 13 2 2 4 2 6 2 a - a + a + a z + 4 a z - 2 a z + a z + 2 a z - 4 a z - 4 a z - 8 2 10 2 12 2 3 3 7 3 9 3 11 3 > a z - a z + 2 a z + 7 a z - 8 a z + 6 a z + 5 a z - 13 3 2 4 4 4 6 4 8 4 10 4 12 4 > 2 a z - 3 a z + 8 a z + 11 a z + 6 a z + a z - 5 a z - 3 5 5 5 7 5 9 5 11 5 13 5 2 6 > 9 a z - 2 a z + 6 a z - 10 a z - 8 a z + a z + a z - 4 6 6 6 8 6 10 6 12 6 3 7 5 7 > 10 a z - 15 a z - 12 a z - 5 a z + 3 a z + 3 a z - 3 a z - 7 7 9 7 11 7 4 8 6 8 8 8 10 8 > 8 a z + 3 a z + 5 a z + 4 a z + 6 a z + 7 a z + 5 a z + 5 9 7 9 9 9 6 10 8 10 > 3 a z + 6 a z + 3 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 208]], Vassiliev[3][Knot[11, Alternating, 208]]}
Out[15]=	{6, -15}
In[16]:=	Kh[Knot[11, Alternating, 208]][q, t]
Out[16]=	3 4 1 2 1 5 2 6 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 23 9 21 8 19 8 19 7 17 7 17 6 15 6 q q q t q t q t q t q t q t q t 9 6 8 9 8 8 7 8 3 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 15 5 13 5 13 4 11 4 11 3 9 3 9 2 7 2 7 q t q t q t q t q t q t q t q t q t 7 t 2 t 2 > ---- + -- + --- + q t 5 3 q q t q

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a208

K11a207 K11a209