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

K11a216 K11a218

Knotscape

This page is passe. Go here instead!    The Knot K11a217 Visit K11a217's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 12 18 14 20 16 2 8 22 6 10

Alexander Polynomial:

- t-4 + 6t-3 - 16t-2 + 29t-1 - 35 + 29t - 16t2 + 6t3 - t4

Conway Polynomial:

1 + 3z2 - 2z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{139, -2}

Jones Polynomial:

- q-8 + 3q-7 - 8q-6 + 14q-5 - 19q-4 + 23q-3 - 22q-2 + 20q-1 - 15 + 9q - 4q2 + q3

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

{...}

A2 (sl(3)) Invariant:

- q-24 - q-20 - 3q-18 + 4q-16 - 2q-14 + 3q-12 + 3q-10 - 2q-8 + 5q-6 - 5q-4 + 3q-2 - 1 - 2q2 + 3q4 - 2q6 + q8

HOMFLY-PT Polynomial:

1 + 3z2 + 3z4 + z6 - 3a2 - 9a2z2 - 10a2z4 - 5a2z6 - a2z8 + 6a4 + 12a4z2 + 8a4z4 + 2a4z6 - 3a6 - 3a6z2 - a6z4

Kauffman Polynomial:

a-2z2 - 2a-2z4 + a-2z6 - a-1z + 6a-1z3 - 9a-1z5 + 4a-1z7 + 1 - 3z2 + 9z4 - 15z6 + 7z8 - 3az + 10az3 - 11az5 - 5az7 + 6az9 + 3a2 - 14a2z2 + 31a2z4 - 38a2z6 + 13a2z8 + 2a2z10 - 5a3z + 13a3z3 - 5a3z5 - 16a3z7 + 12a3z9 + 6a4 - 20a4z2 + 38a4z4 - 38a4z6 + 14a4z8 + 2a4z10 - 7a5z + 18a5z3 - 13a5z5 - a5z7 + 6a5z9 + 3a6 - 9a6z2 + 14a6z4 - 13a6z6 + 8a6z8 - 3a7z + 7a7z3 - 9a7z5 + 6a7z7 + a8z2 - 4a8z4 + 3a8z6 + a9z - 2a9z3 + a9z5

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

{3, -6}

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 11217. 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                   6 1   j = 1                 9 3     j = -1               11 6       j = -3             12 10         j = -5           11 10           j = -7         8 12             j = -9       6 11               j = -11     2 8                 j = -13   1 6                   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, 217]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 217]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 3, 13, 4], X[18, 5, 19, 6], X[14, 8, 15, 7], > X[20, 9, 21, 10], X[16, 12, 17, 11], X[2, 13, 3, 14], X[8, 16, 9, 15], > X[22, 17, 1, 18], X[6, 19, 7, 20], X[10, 21, 11, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 217]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -10, 4, -8, 5, -11, 6, -2, 7, -4, 8, -6, 9, -3, 10, > -5, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 217]]
Out[5]=	DTCode[4, 12, 18, 14, 20, 16, 2, 8, 22, 6, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 217]][t]
Out[6]=	-4 6 16 29 2 3 4 -35 - t + -- - -- + -- + 29 t - 16 t + 6 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 217]][z]
Out[7]=	2 6 8 1 + 3 z - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 217]}
In[9]:=	{KnotDet[Knot[11, Alternating, 217]], KnotSignature[Knot[11, Alternating, 217]]}
Out[9]=	{139, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 217]][q]
Out[10]=	-8 3 8 14 19 23 22 20 2 3 -15 - q + -- - -- + -- - -- + -- - -- + -- + 9 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, 217]}
In[12]:=	A2Invariant[Knot[11, Alternating, 217]][q]
Out[12]=	-24 -20 3 4 2 3 3 2 5 5 3 2 -1 - q - q - --- + --- - --- + --- + --- - -- + -- - -- + -- - 2 q + 18 16 14 12 10 8 6 4 2 q q q q q q q q q 4 6 8 > 3 q - 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 217]][a, z]
Out[13]=	2 4 6 2 2 2 4 2 6 2 4 1 - 3 a + 6 a - 3 a + 3 z - 9 a z + 12 a z - 3 a z + 3 z - 2 4 4 4 6 4 6 2 6 4 6 2 8 > 10 a z + 8 a z - a z + z - 5 a z + 2 a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 217]][a, z]
Out[14]=	2 4 6 z 3 5 7 9 2 1 + 3 a + 6 a + 3 a - - - 3 a z - 5 a z - 7 a z - 3 a z + a z - 3 z + a 2 3 z 2 2 4 2 6 2 8 2 6 z 3 3 3 > -- - 14 a z - 20 a z - 9 a z + a z + ---- + 10 a z + 13 a z + 2 a a 4 5 3 7 3 9 3 4 2 z 2 4 4 4 > 18 a z + 7 a z - 2 a z + 9 z - ---- + 31 a z + 38 a z + 2 a 5 6 4 8 4 9 z 5 3 5 5 5 7 5 > 14 a z - 4 a z - ---- - 11 a z - 5 a z - 13 a z - 9 a z + a 6 7 9 5 6 z 2 6 4 6 6 6 8 6 4 z > a z - 15 z + -- - 38 a z - 38 a z - 13 a z + 3 a z + ---- - 2 a a 7 3 7 5 7 7 7 8 2 8 4 8 > 5 a z - 16 a z - a z + 6 a z + 7 z + 13 a z + 14 a z + 6 8 9 3 9 5 9 2 10 4 10 > 8 a z + 6 a z + 12 a z + 6 a z + 2 a z + 2 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 217]], Vassiliev[3][Knot[11, Alternating, 217]]}
Out[15]=	{3, -6}
In[16]:=	Kh[Knot[11, Alternating, 217]][q, t]
Out[16]=	10 11 1 2 1 6 2 8 6 11 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 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 8 12 11 10 12 6 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 9 q t + 3 q t + 6 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 K11a217

K11a216 K11a218