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

K11a244 K11a246

Knotscape

This page is passe. Go here instead!    The Knot K11a245 Visit K11a245's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 14 20 22 16 18 2 12 10 8 6

Alexander Polynomial:

3t-3 - 9t-2 + 16t-1 - 19 + 16t - 9t2 + 3t3

Conway Polynomial:

1 + 7z2 + 9z4 + 3z6

Other knots with the same Alexander/Conway Polynomial:

{1066, ...}

Determinant and Signature:

{75, 6}

Jones Polynomial:

q3 - q4 + 4q5 - 6q6 + 9q7 - 12q8 + 12q9 - 11q10 + 9q11 - 6q12 + 3q13 - q14

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

{...}

A2 (sl(3)) Invariant:

q10 + 3q14 + q16 + q18 + 2q20 - 3q22 - 2q26 - q28 + 2q30 - q32 + 2q34 - q36 - q38 + q40 - q42

HOMFLY-PT Polynomial:

- a-12 - 2a-12z2 - a-12z4 + 2a-10 + 3a-10z2 + 3a-10z4 + a-10z6 - 4a-8 - 2a-8z2 + 2a-8z4 + a-8z6 + 4a-6 + 8a-6z2 + 5a-6z4 + a-6z6

Kauffman Polynomial:

- 2a-17z3 + a-17z5 + a-16z2 - 6a-16z4 + 3a-16z6 - 2a-15z + 8a-15z3 - 12a-15z5 + 5a-15z7 - 5a-14z2 + 14a-14z4 - 13a-14z6 + 5a-14z8 + a-13z - 2a-13z3 + 7a-13z5 - 6a-13z7 + 3a-13z9 - a-12 - a-12z2 + 7a-12z4 - 4a-12z6 + a-12z8 + a-12z10 + 5a-11z - 20a-11z3 + 25a-11z5 - 13a-11z7 + 4a-11z9 - 2a-10 + 6a-10z2 - 15a-10z4 + 11a-10z6 - 3a-10z8 + a-10z10 + 6a-9z - 10a-9z3 + 3a-9z5 - a-9z7 + a-9z9 - 4a-8 + 9a-8z2 - 7a-8z4 + a-8z8 + 4a-7z - 2a-7z3 - 2a-7z5 + a-7z7 - 4a-6 + 8a-6z2 - 5a-6z4 + a-6z6

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

{7, 19}

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=6 is the signature of 11245. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 r = 10 r = 11 j = 29                       1 j = 27                     2   j = 25                   4 1   j = 23                 5 2     j = 21               6 4       j = 19             6 5         j = 17           6 6           j = 15         3 6             j = 13       3 6               j = 11     1 3                 j = 9     3                   j = 7 1 1                     j = 5 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, 245]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 245]]
Out[3]=	PD[X[4, 2, 5, 1], X[14, 4, 15, 3], X[20, 6, 21, 5], X[22, 8, 1, 7], > X[16, 10, 17, 9], X[18, 12, 19, 11], X[2, 14, 3, 13], X[12, 16, 13, 15], > X[10, 18, 11, 17], X[8, 20, 9, 19], X[6, 22, 7, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 245]]
Out[4]=	GaussCode[1, -7, 2, -1, 3, -11, 4, -10, 5, -9, 6, -8, 7, -2, 8, -5, 9, -6, 10, > -3, 11, -4]
In[5]:=	DTCode[Knot[11, Alternating, 245]]
Out[5]=	DTCode[4, 14, 20, 22, 16, 18, 2, 12, 10, 8, 6]
In[6]:=	alex = Alexander[Knot[11, Alternating, 245]][t]
Out[6]=	3 9 16 2 3 -19 + -- - -- + -- + 16 t - 9 t + 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 245]][z]
Out[7]=	2 4 6 1 + 7 z + 9 z + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 66], Knot[11, Alternating, 245]}
In[9]:=	{KnotDet[Knot[11, Alternating, 245]], KnotSignature[Knot[11, Alternating, 245]]}
Out[9]=	{75, 6}
In[10]:=	J=Jones[Knot[11, Alternating, 245]][q]
Out[10]=	3 4 5 6 7 8 9 10 11 12 13 q - q + 4 q - 6 q + 9 q - 12 q + 12 q - 11 q + 9 q - 6 q + 3 q - 14 > q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 245]}
In[12]:=	A2Invariant[Knot[11, Alternating, 245]][q]
Out[12]=	10 14 16 18 20 22 26 28 30 32 34 q + 3 q + q + q + 2 q - 3 q - 2 q - q + 2 q - q + 2 q - 36 38 40 42 > q - q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 245]][a, z]
Out[13]=	2 2 2 2 4 4 4 4 -12 2 4 4 2 z 3 z 2 z 8 z z 3 z 2 z 5 z -a + --- - -- + -- - ---- + ---- - ---- + ---- - --- + ---- + ---- + ---- + 10 8 6 12 10 8 6 12 10 8 6 a a a a a a a a a a a 6 6 6 z z z > --- + -- + -- 10 8 6 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 245]][a, z]
Out[14]=	2 2 2 2 -12 2 4 4 2 z z 5 z 6 z 4 z z 5 z z 6 z -a - --- - -- - -- - --- + --- + --- + --- + --- + --- - ---- - --- + ---- + 10 8 6 15 13 11 9 7 16 14 12 10 a a a a a a a a a a a a 2 2 3 3 3 3 3 3 4 4 9 z 8 z 2 z 8 z 2 z 20 z 10 z 2 z 6 z 14 z > ---- + ---- - ---- + ---- - ---- - ----- - ----- - ---- - ---- + ----- + 8 6 17 15 13 11 9 7 16 14 a a a a a a a a a a 4 4 4 4 5 5 5 5 5 5 7 z 15 z 7 z 5 z z 12 z 7 z 25 z 3 z 2 z > ---- - ----- - ---- - ---- + --- - ----- + ---- + ----- + ---- - ---- + 12 10 8 6 17 15 13 11 9 7 a a a a a a a a a a 6 6 6 6 6 7 7 7 7 7 8 3 z 13 z 4 z 11 z z 5 z 6 z 13 z z z 5 z > ---- - ----- - ---- + ----- + -- + ---- - ---- - ----- - -- + -- + ---- + 16 14 12 10 6 15 13 11 9 7 14 a a a a a a a a a a a 8 8 8 9 9 9 10 10 z 3 z z 3 z 4 z z z z > --- - ---- + -- + ---- + ---- + -- + --- + --- 12 10 8 13 11 9 12 10 a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 245]], Vassiliev[3][Knot[11, Alternating, 245]]}
Out[15]=	{7, 19}
In[16]:=	Kh[Knot[11, Alternating, 245]][q, t]
Out[16]=	5 7 7 9 2 11 2 11 3 13 3 13 4 15 4 q + q + q t + 3 q t + q t + 3 q t + 3 q t + 6 q t + 3 q t + 15 5 17 5 17 6 19 6 19 7 21 7 > 6 q t + 6 q t + 6 q t + 6 q t + 5 q t + 6 q t + 21 8 23 8 23 9 25 9 25 10 27 10 29 11 > 4 q t + 5 q t + 2 q t + 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a245

K11a244 K11a246