K11a27

K11a26

K11a28

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a27 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8394 X12,6,13,5 X14,7,15,8 X2,9,3,10 X18,11,19,12 X22,14,1,13 X20,15,21,16 X10,17,11,18 X16,19,17,20 X6,21,7,22
Gauss code	1, -5, 2, -1, 3, -11, 4, -2, 5, -9, 6, -3, 7, -4, 8, -10, 9, -6, 10, -8, 11, -7
Dowker-Thistlethwaite code	4 8 12 14 2 18 22 20 10 16 6

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a27/ThurstonBennequinNumber Hyperbolic Volume 17.0192 A-Polynomial See Data:K11a27/A-polynomial

[edit Notes for K11a27's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant 2

[edit Notes for K11a27's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-13t^{2}+34t-45+34t^{-1}-13t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}-z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 143, -2 }
Jones polynomial	$-q^{2}+4q-8+15q^{-1}-20q^{-2}+23q^{-3}-23q^{-4}+20q^{-5}-15q^{-6}+9q^{-7}-4q^{-8}+q^{-9}$
HOMFLY-PT polynomial (db, data sources)	$z^{2}a^{8}-2z^{4}a^{6}-z^{2}a^{6}+a^{6}+z^{6}a^{4}-3z^{2}a^{4}-3a^{4}+z^{6}a^{2}+2z^{4}a^{2}+4z^{2}a^{2}+3a^{2}-z^{4}-z^{2}$
Kauffman polynomial (db, data sources)	$z^{6}a^{10}-2z^{4}a^{10}+z^{2}a^{10}+4z^{7}a^{9}-9z^{5}a^{9}+6z^{3}a^{9}-za^{9}+7z^{8}a^{8}-15z^{6}a^{8}+9z^{4}a^{8}-2z^{2}a^{8}+6z^{9}a^{7}-5z^{7}a^{7}-10z^{5}a^{7}+9z^{3}a^{7}-2za^{7}+2z^{10}a^{6}+13z^{8}a^{6}-35z^{6}a^{6}+22z^{4}a^{6}-3z^{2}a^{6}-a^{6}+12z^{9}a^{5}-14z^{7}a^{5}-7z^{5}a^{5}+10z^{3}a^{5}-2za^{5}+2z^{10}a^{4}+14z^{8}a^{4}-29z^{6}a^{4}+11z^{4}a^{4}+6z^{2}a^{4}-3a^{4}+6z^{9}a^{3}+2z^{7}a^{3}-16z^{5}a^{3}+12z^{3}a^{3}-2za^{3}+8z^{8}a^{2}-6z^{6}a^{2}-6z^{4}a^{2}+9z^{2}a^{2}-3a^{2}+7z^{7}a-9z^{5}a+4z^{3}a-za+4z^{6}-6z^{4}+3z^{2}+z^{5}a^{-1}-z^{3}a^{-1}$
The A2 invariant	Data:K11a27/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a27/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a27 Quantum Invariants.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a27"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$2t^{3}-13t^{2}+34t-45+34t^{-1}-13t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$2z^{6}-z^{4}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 143, -2 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$-q^{2}+4q-8+15q^{-1}-20q^{-2}+23q^{-3}-23q^{-4}+20q^{-5}-15q^{-6}+9q^{-7}-4q^{-8}+q^{-9}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$z^{2}a^{8}-2z^{4}a^{6}-z^{2}a^{6}+a^{6}+z^{6}a^{4}-3z^{2}a^{4}-3a^{4}+z^{6}a^{2}+2z^{4}a^{2}+4z^{2}a^{2}+3a^{2}-z^{4}-z^{2}$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$z^{6}a^{10}-2z^{4}a^{10}+z^{2}a^{10}+4z^{7}a^{9}-9z^{5}a^{9}+6z^{3}a^{9}-za^{9}+7z^{8}a^{8}-15z^{6}a^{8}+9z^{4}a^{8}-2z^{2}a^{8}+6z^{9}a^{7}-5z^{7}a^{7}-10z^{5}a^{7}+9z^{3}a^{7}-2za^{7}+2z^{10}a^{6}+13z^{8}a^{6}-35z^{6}a^{6}+22z^{4}a^{6}-3z^{2}a^{6}-a^{6}+12z^{9}a^{5}-14z^{7}a^{5}-7z^{5}a^{5}+10z^{3}a^{5}-2za^{5}+2z^{10}a^{4}+14z^{8}a^{4}-29z^{6}a^{4}+11z^{4}a^{4}+6z^{2}a^{4}-3a^{4}+6z^{9}a^{3}+2z^{7}a^{3}-16z^{5}a^{3}+12z^{3}a^{3}-2za^{3}+8z^{8}a^{2}-6z^{6}a^{2}-6z^{4}a^{2}+9z^{2}a^{2}-3a^{2}+7z^{7}a-9z^{5}a+4z^{3}a-za+4z^{6}-6z^{4}+3z^{2}+z^{5}a^{-1}-z^{3}a^{-1}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a27"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ $2t^{3}-13t^{2}+34t-45+34t^{-1}-13t^{-2}+2t^{-3}$, $-q^{2}+4q-8+15q^{-1}-20q^{-2}+23q^{-3}-23q^{-4}+20q^{-5}-15q^{-6}+9q^{-7}-4q^{-8}+q^{-9}$ }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(0, 1)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $0$ $8$ $0$ $-16$ $8$ $0$ $-{\frac {112}{3}}$ $-{\frac {160}{3}}$ $-24$ $0$ $32$ $0$ $0$ $296$ $-{\frac {152}{3}}$ ${\frac {1096}{3}}$ $-{\frac {8}{3}}$ $56$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). 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 K11a27. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 5                       1 -1 3                     3   3 1                   5 1   -4 -1                 10 3     7 -3               11 6       -5 -5             12 9         3 -7           11 11           0 -9         9 12             -3 -11       6 11               5 -13     3 9                 -6 -15   1 6                   5 -17   3                     -3 -19 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-8$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-5$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-4$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-3$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=-2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=0$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages. See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate). See/edit the Hoste-Thistlethwaite_Splice_Base (expert). Back to the top.

K11a26 K11a28