# 9 1

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 9 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 1 at Knotilus! 9_1 should perhaps be called "The Nonafoil Knot", following the trefoil knot, the cinquefoil knot and (maybe) the septafoil knot. The next in the series is K11a367. See also T(9,2).

 Interlaced form of 9/2 star polygon or "nonagram" Decorative interlaced form of 9/2 star polygon or "nonagram" Alternate interlaced form of 9/2 star polygon or "nonagram"

### Knot presentations

 Planar diagram presentation X1,10,2,11 X3,12,4,13 X5,14,6,15 X7,16,8,17 X9,18,10,1 X11,2,12,3 X13,4,14,5 X15,6,16,7 X17,8,18,9 Gauss code -1, 6, -2, 7, -3, 8, -4, 9, -5, 1, -6, 2, -7, 3, -8, 4, -9, 5 Dowker-Thistlethwaite code 10 12 14 16 18 2 4 6 8 Conway Notation [9]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 9, width is 2,

Braid index is 2

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

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 4 3-genus 4 Bridge index 2 Super bridge index 4 Nakanishi index 1 Maximal Thurston-Bennequin number [-18][7] Hyperbolic Volume Not hyperbolic A-Polynomial See Data:9 1/A-polynomial

### Four dimensional invariants

 Smooth 4 genus ${\displaystyle 4}$ Topological 4 genus ${\displaystyle 4}$ Concordance genus ${\displaystyle 4}$ Rasmussen s-Invariant -8

### Polynomial invariants

 Alexander polynomial ${\displaystyle t^{4}-t^{3}+t^{2}-t+1-t^{-1}+t^{-2}-t^{-3}+t^{-4}}$ Conway polynomial ${\displaystyle z^{8}+7z^{6}+15z^{4}+10z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 9, -8 } Jones polynomial ${\displaystyle q^{-4}+q^{-6}-q^{-7}+q^{-8}-q^{-9}+q^{-10}-q^{-11}+q^{-12}-q^{-13}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle -z^{6}a^{10}-6z^{4}a^{10}-10z^{2}a^{10}-4a^{10}+z^{8}a^{8}+8z^{6}a^{8}+21z^{4}a^{8}+20z^{2}a^{8}+5a^{8}}$ Kauffman polynomial (db, data sources) ${\displaystyle za^{17}+z^{2}a^{16}+z^{3}a^{15}-za^{15}+z^{4}a^{14}-2z^{2}a^{14}+z^{5}a^{13}-3z^{3}a^{13}+za^{13}+z^{6}a^{12}-4z^{4}a^{12}+3z^{2}a^{12}+z^{7}a^{11}-5z^{5}a^{11}+6z^{3}a^{11}-za^{11}+z^{8}a^{10}-7z^{6}a^{10}+16z^{4}a^{10}-14z^{2}a^{10}+4a^{10}+z^{7}a^{9}-6z^{5}a^{9}+10z^{3}a^{9}-4za^{9}+z^{8}a^{8}-8z^{6}a^{8}+21z^{4}a^{8}-20z^{2}a^{8}+5a^{8}}$ The A2 invariant ${\displaystyle -q^{38}-q^{36}-q^{34}+q^{22}+q^{20}+2q^{18}+q^{16}+q^{14}}$ The G2 invariant ${\displaystyle q^{216}-q^{172}-q^{170}-q^{164}-q^{162}-q^{160}-q^{154}-q^{152}-q^{126}-q^{120}-q^{118}-q^{116}-q^{114}-q^{108}+q^{100}+q^{98}+q^{94}+2q^{92}+2q^{90}+2q^{88}+q^{86}+q^{84}+2q^{82}+2q^{80}+q^{78}+q^{74}+q^{72}+q^{70}}$

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

Same Alexander/Conway Polynomial: {}

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

### Vassiliev invariants

 V2 and V3: (10, -30)
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 ${\displaystyle 40}$ ${\displaystyle -240}$ ${\displaystyle 800}$ ${\displaystyle {\frac {5660}{3}}}$ ${\displaystyle {\frac {820}{3}}}$ ${\displaystyle -9600}$ ${\displaystyle -16448}$ ${\displaystyle -2912}$ ${\displaystyle -1968}$ ${\displaystyle {\frac {32000}{3}}}$ ${\displaystyle 28800}$ ${\displaystyle {\frac {226400}{3}}}$ ${\displaystyle {\frac {32800}{3}}}$ ${\displaystyle {\frac {440263}{3}}}$ ${\displaystyle {\frac {20516}{3}}}$ ${\displaystyle {\frac {474988}{9}}}$ ${\displaystyle {\frac {6637}{9}}}$ ${\displaystyle {\frac {19927}{3}}}$

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 ${\displaystyle s=}$-8 is the signature of 9 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-9-8-7-6-5-4-3-2-10χ
-7         11
-9         11
-11       1  1
-13          0
-15     11   0
-17          0
-19   11     0
-21          0
-23 11       0
-25          0
-271         -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-9}$ ${\displaystyle i=-7}$ ${\displaystyle r=-9}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-8}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-7}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$