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

K11a278 K11a280

Knotscape

This page is passe. Go here instead!    The Knot K11a279 Visit K11a279's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X6271 X10,3,11,4 X16,5,17,6 X12,8,13,7 X18,9,19,10 X2,11,3,12 X20,14,21,13 X22,16,1,15 X4,17,5,18 X8,19,9,20 X14,22,15,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 10 16 12 18 2 20 22 4 8 14

Alexander Polynomial:

3t-3 - 12t-2 + 20t-1 - 21 + 20t - 12t2 + 3t3

Conway Polynomial:

1 - z2 + 6z4 + 3z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{91, 2}

Jones Polynomial:

q-5 - 3q-4 + 6q-3 - 9q-2 + 12q-1 - 14 + 14q - 12q2 + 10q3 - 6q4 + 3q5 - q6

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

{...}

A2 (sl(3)) Invariant:

q-16 + q-14 - q-12 + q-10 - 2q-8 - q-6 + q-4 - 2q-2 + 3 + q4 + 2q6 - 2q8 + 2q10 - q12 + q16 - q18

HOMFLY-PT Polynomial:

- 2a-4z2 - a-4z4 - a-2 - a-2z2 + 2a-2z4 + a-2z6 + 5 + 10z2 + 8z4 + 2z6 - 5a2 - 9a2z2 - 3a2z4 + 2a4 + a4z2

Kauffman Polynomial:

a-7z3 + 3a-6z4 - 3a-5z3 + 6a-5z5 + 6a-4z2 - 15a-4z4 + 10a-4z6 - 2a-3z + 12a-3z3 - 25a-3z5 + 12a-3z7 + a-2 - 4a-2z2 + 14a-2z4 - 27a-2z6 + 11a-2z8 - 3a-1z + 11a-1z3 - 4a-1z5 - 15a-1z7 + 7a-1z9 + 5 - 31z2 + 67z4 - 50z6 + 7z8 + 2z10 + az - 18az3 + 51az5 - 42az7 + 10az9 + 5a2 - 28a2z2 + 44a2z4 - 18a2z6 - 3a2z8 + 2a2z10 + 2a3z - 13a3z3 + 24a3z5 - 15a3z7 + 3a3z9 + 2a4 - 7a4z2 + 9a4z4 - 5a4z6 + a4z8

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=2 is the signature of 11279. 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 = 13                       1 j = 11                     2   j = 9                   4 1   j = 7                 6 2     j = 5               6 4       j = 3             8 6         j = 1           7 7           j = -1         5 7             j = -3       4 7               j = -5     2 5                 j = -7   1 4                   j = -9   2                     j = -11 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, 279]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 279]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 3, 11, 4], X[16, 5, 17, 6], X[12, 8, 13, 7], > X[18, 9, 19, 10], X[2, 11, 3, 12], X[20, 14, 21, 13], X[22, 16, 1, 15], > X[4, 17, 5, 18], X[8, 19, 9, 20], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 279]]
Out[4]=	GaussCode[1, -6, 2, -9, 3, -1, 4, -10, 5, -2, 6, -4, 7, -11, 8, -3, 9, -5, 10, > -7, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 279]]
Out[5]=	DTCode[6, 10, 16, 12, 18, 2, 20, 22, 4, 8, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 279]][t]
Out[6]=	3 12 20 2 3 -21 + -- - -- + -- + 20 t - 12 t + 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 279]][z]
Out[7]=	2 4 6 1 - z + 6 z + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 279]}
In[9]:=	{KnotDet[Knot[11, Alternating, 279]], KnotSignature[Knot[11, Alternating, 279]]}
Out[9]=	{91, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 279]][q]
Out[10]=	-5 3 6 9 12 2 3 4 5 6 -14 + q - -- + -- - -- + -- + 14 q - 12 q + 10 q - 6 q + 3 q - q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 279]}
In[12]:=	A2Invariant[Knot[11, Alternating, 279]][q]
Out[12]=	-16 -14 -12 -10 2 -6 -4 2 4 6 8 3 + q + q - q + q - -- - q + q - -- + q + 2 q - 2 q + 8 2 q q 10 12 16 18 > 2 q - q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 279]][a, z]
Out[13]=	2 2 4 -2 2 4 2 2 z z 2 2 4 2 4 z 5 - a - 5 a + 2 a + 10 z - ---- - -- - 9 a z + a z + 8 z - -- + 4 2 4 a a a 4 6 2 z 2 4 6 z > ---- - 3 a z + 2 z + -- 2 2 a a
In[14]:=	Kauffman[Knot[11, Alternating, 279]][a, z]
Out[14]=	2 2 -2 2 4 2 z 3 z 3 2 6 z 4 z 5 + a + 5 a + 2 a - --- - --- + a z + 2 a z - 31 z + ---- - ---- - 3 a 4 2 a a a 3 3 3 3 2 2 4 2 z 3 z 12 z 11 z 3 3 3 > 28 a z - 7 a z + -- - ---- + ----- + ----- - 18 a z - 13 a z + 7 5 3 a a a a 4 4 4 5 5 5 4 3 z 15 z 14 z 2 4 4 4 6 z 25 z 4 z > 67 z + ---- - ----- + ----- + 44 a z + 9 a z + ---- - ----- - ---- + 6 4 2 5 3 a a a a a a 6 6 7 5 3 5 6 10 z 27 z 2 6 4 6 12 z > 51 a z + 24 a z - 50 z + ----- - ----- - 18 a z - 5 a z + ----- - 4 2 3 a a a 7 8 9 15 z 7 3 7 8 11 z 2 8 4 8 7 z > ----- - 42 a z - 15 a z + 7 z + ----- - 3 a z + a z + ---- + a 2 a a 9 3 9 10 2 10 > 10 a z + 3 a z + 2 z + 2 a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 279]], Vassiliev[3][Knot[11, Alternating, 279]]}
Out[15]=	{-1, 2}
In[16]:=	Kh[Knot[11, Alternating, 279]][q, t]
Out[16]=	3 1 2 1 4 2 5 4 7 7 q + 8 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ----- + 11 6 9 5 7 5 7 4 5 4 5 3 3 3 3 2 q t q t q t q t q t q t q t q t 5 7 7 q 3 5 5 2 7 2 7 3 > ---- + --- + --- + 6 q t + 6 q t + 4 q t + 6 q t + 2 q t + 2 q t t q t 9 3 9 4 11 4 13 5 > 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a279

K11a278 K11a280