# T(11,2)

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 See other torus knots Visit T(11,2) at Knotilus! Edit T(11,2) Quick Notes See also K11a367.

### Knot presentations

 Planar diagram presentation X5,17,6,16 X17,7,18,6 X7,19,8,18 X19,9,20,8 X9,21,10,20 X21,11,22,10 X11,1,12,22 X1,13,2,12 X13,3,14,2 X3,15,4,14 X15,5,16,4 Gauss code -8, 9, -10, 11, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 1, -2, 3, -4, 5, -6, 7 Dowker-Thistlethwaite code 12 14 16 18 20 22 2 4 6 8 10

### Polynomial invariants

 Alexander polynomial ${\displaystyle t^{5}-t^{4}+t^{3}-t^{2}+t-1+t^{-1}-t^{-2}+t^{-3}-t^{-4}+t^{-5}}$ Conway polynomial ${\displaystyle z^{10}+9z^{8}+28z^{6}+35z^{4}+15z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 11, 10 } Jones polynomial ${\displaystyle -q^{16}+q^{15}-q^{14}+q^{13}-q^{12}+q^{11}-q^{10}+q^{9}-q^{8}+q^{7}+q^{5}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle z^{10}a^{-10}+10z^{8}a^{-10}-z^{8}a^{-12}+36z^{6}a^{-10}-8z^{6}a^{-12}+56z^{4}a^{-10}-21z^{4}a^{-12}+35z^{2}a^{-10}-20z^{2}a^{-12}+6a^{-10}-5a^{-12}}$ Kauffman polynomial (db, data sources) ${\displaystyle z^{10}a^{-10}+z^{10}a^{-12}+z^{9}a^{-11}+z^{9}a^{-13}-10z^{8}a^{-10}-9z^{8}a^{-12}+z^{8}a^{-14}-8z^{7}a^{-11}-7z^{7}a^{-13}+z^{7}a^{-15}+36z^{6}a^{-10}+29z^{6}a^{-12}-6z^{6}a^{-14}+z^{6}a^{-16}+21z^{5}a^{-11}+15z^{5}a^{-13}-5z^{5}a^{-15}+z^{5}a^{-17}-56z^{4}a^{-10}-41z^{4}a^{-12}+10z^{4}a^{-14}-4z^{4}a^{-16}+z^{4}a^{-18}-20z^{3}a^{-11}-10z^{3}a^{-13}+6z^{3}a^{-15}-3z^{3}a^{-17}+z^{3}a^{-19}+35z^{2}a^{-10}+25z^{2}a^{-12}-4z^{2}a^{-14}+3z^{2}a^{-16}-2z^{2}a^{-18}+z^{2}a^{-20}+5za^{-11}+za^{-13}-za^{-15}+za^{-17}-za^{-19}+za^{-21}-6a^{-10}-5a^{-12}}$ The A2 invariant Data:T(11,2)/QuantumInvariant/A2/1,0 The G2 invariant Data:T(11,2)/QuantumInvariant/G2/1,0

### "Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a367,}

Same Jones Polynomial (up to mirroring, ${\displaystyle q\leftrightarrow q^{-1}}$): {K11a367,}

### Vassiliev invariants

 V2 and V3: (15, 55)
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 Data:T(11,2)/V 2,1 Data:T(11,2)/V 3,1 Data:T(11,2)/V 4,1 Data:T(11,2)/V 4,2 Data:T(11,2)/V 4,3 Data:T(11,2)/V 5,1 Data:T(11,2)/V 5,2 Data:T(11,2)/V 5,3 Data:T(11,2)/V 5,4 Data:T(11,2)/V 6,1 Data:T(11,2)/V 6,2 Data:T(11,2)/V 6,3 Data:T(11,2)/V 6,4 Data:T(11,2)/V 6,5 Data:T(11,2)/V 6,6 Data:T(11,2)/V 6,7 Data:T(11,2)/V 6,8 Data:T(11,2)/V 6,9

V2,1 through V6,9 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 ${\displaystyle t^{r}q^{j}}$ are shown, along with their alternating sums ${\displaystyle \chi }$ (fixed ${\displaystyle j}$, alternation over ${\displaystyle r}$). The squares with yellow highlighting are those on the "critical diagonals", where ${\displaystyle j-2r=s+1}$ or ${\displaystyle j-2r=s-1}$, where ${\displaystyle s=}$10 is the signature of T(11,2). Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
01234567891011χ
33           1-1
31            0
29         11 0
27            0
25       11   0
23            0
21     11     0
19            0
17   11       0
15            0
13  1         1
111           1
91           1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=9}$ ${\displaystyle i=11}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=1}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=7}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=8}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=9}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=10}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=11}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$

### 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 Torus Knot Page master template (intermediate).

See/edit the Torus Knot_Splice_Base (expert).

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