K11n27

K11n26

K11n28

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n27 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X5,13,6,12 X2837 X9,15,10,14 X11,18,12,19 X13,7,14,6 X15,21,16,20 X17,1,18,22 X19,10,20,11 X21,17,22,16
Gauss code	1, -4, 2, -1, -3, 7, 4, -2, -5, 10, -6, 3, -7, 5, -8, 11, -9, 6, -10, 8, -11, 9
Dowker-Thistlethwaite code	4 8 -12 2 -14 -18 -6 -20 -22 -10 -16

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number 3 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n27/ThurstonBennequinNumber Hyperbolic Volume 9.33368 A-Polynomial See Data:K11n27/A-polynomial

[edit Notes for K11n27's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $4$ Rasmussen s-Invariant -6

[edit Notes for K11n27's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^4+3 t^3-3 t^2+2 t-1+2 t^{-1} -3 t^{-2} +3 t^{-3} - t^{-4}$
Conway polynomial	$-z^8-5 z^6-5 z^4+z^2+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 19, 6 }
Jones polynomial	$q^9-2 q^8+2 q^7-3 q^6+3 q^5-3 q^4+3 q^3-q^2+q$
HOMFLY-PT polynomial (db, data sources)	$-z^8 a^{-6} +z^6 a^{-4} -7 z^6 a^{-6} +z^6 a^{-8} +6 z^4 a^{-4} -17 z^4 a^{-6} +6 z^4 a^{-8} +11 z^2 a^{-4} -19 z^2 a^{-6} +10 z^2 a^{-8} -z^2 a^{-10} +6 a^{-4} -9 a^{-6} +5 a^{-8} - a^{-10}$
Kauffman polynomial (db, data sources)	$z^9 a^{-5} +z^9 a^{-7} +z^8 a^{-4} +4 z^8 a^{-6} +3 z^8 a^{-8} -5 z^7 a^{-5} -2 z^7 a^{-7} +3 z^7 a^{-9} -7 z^6 a^{-4} -24 z^6 a^{-6} -16 z^6 a^{-8} +z^6 a^{-10} +5 z^5 a^{-5} -10 z^5 a^{-7} -15 z^5 a^{-9} +17 z^4 a^{-4} +45 z^4 a^{-6} +24 z^4 a^{-8} -4 z^4 a^{-10} +4 z^3 a^{-5} +22 z^3 a^{-7} +18 z^3 a^{-9} -17 z^2 a^{-4} -33 z^2 a^{-6} -15 z^2 a^{-8} +z^2 a^{-10} -5 z a^{-5} -10 z a^{-7} -6 z a^{-9} -z a^{-11} +6 a^{-4} +9 a^{-6} +5 a^{-8} + a^{-10}$
The A2 invariant	$q^{-4} + q^{-6} +2 q^{-8} +2 q^{-10} + q^{-12} -2 q^{-16} - q^{-18} -3 q^{-20} + q^{-26} + q^{-28} + q^{-32} - q^{-34}$
The G2 invariant	Data:K11n27/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n27 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["K11n27"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^4+3 t^3-3 t^2+2 t-1+2 t^{-1} -3 t^{-2} +3 t^{-3} - t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^8-5 z^6-5 z^4+z^2+1$

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]=

{ 19, 6 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^9-2 q^8+2 q^7-3 q^6+3 q^5-3 q^4+3 q^3-q^2+q$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^8 a^{-6} +z^6 a^{-4} -7 z^6 a^{-6} +z^6 a^{-8} +6 z^4 a^{-4} -17 z^4 a^{-6} +6 z^4 a^{-8} +11 z^2 a^{-4} -19 z^2 a^{-6} +10 z^2 a^{-8} -z^2 a^{-10} +6 a^{-4} -9 a^{-6} +5 a^{-8} - a^{-10}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^9 a^{-5} +z^9 a^{-7} +z^8 a^{-4} +4 z^8 a^{-6} +3 z^8 a^{-8} -5 z^7 a^{-5} -2 z^7 a^{-7} +3 z^7 a^{-9} -7 z^6 a^{-4} -24 z^6 a^{-6} -16 z^6 a^{-8} +z^6 a^{-10} +5 z^5 a^{-5} -10 z^5 a^{-7} -15 z^5 a^{-9} +17 z^4 a^{-4} +45 z^4 a^{-6} +24 z^4 a^{-8} -4 z^4 a^{-10} +4 z^3 a^{-5} +22 z^3 a^{-7} +18 z^3 a^{-9} -17 z^2 a^{-4} -33 z^2 a^{-6} -15 z^2 a^{-8} +z^2 a^{-10} -5 z a^{-5} -10 z a^{-7} -6 z a^{-9} -z a^{-11} +6 a^{-4} +9 a^{-6} +5 a^{-8} + a^{-10}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {10_133,}

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["K11n27"];

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]=

{ $-t^4+3 t^3-3 t^2+2 t-1+2 t^{-1} -3 t^{-2} +3 t^{-3} - t^{-4}$, $q^9-2 q^8+2 q^7-3 q^6+3 q^5-3 q^4+3 q^3-q^2+q$ }

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]=

{10_133,}

Vassiliev invariants

V2 and V3:

(1, 0)

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 $4$ $0$ $8$ $\frac{110}{3}$ $\frac{130}{3}$ $0$ $320$ $160$ $160$ $\frac{32}{3}$ $0$ $\frac{440}{3}$ $\frac{520}{3}$ $\frac{44191}{30}$ $\frac{338}{15}$ $\frac{56462}{45}$ $\frac{2081}{18}$ $\frac{2911}{30}$

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 $t^rq^j$ 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=$6 is the signature of K11n27. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=3$ $i=5$ $i=7$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{3}$ ${\mathbb Z}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

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.

K11n26 K11n28